INTERNATIONAL ATOMIC ENERGY AGENCY, VIENNA, 1974
GLOBAL ANALYSIS AND ITS APPLICATIONS
Vol. II
INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS, TRIESTE
GLOBAL ANALYSIS AND ITS APPLICATIONS
LECTURES PRESENTED AT AN INTERNATIONAL SEMINAR COURSE AT TRIESTE FROM 4 JULY TO 25 AUGUST 1972 ORGANIZED BY THE INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS, TRIESTE
In three volumes
VOL. II
INTERNATIONAL ATOMIC ENERGY AGENCY VIENNA, 1974 THE INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS (ICTP) in Trieste was establishedby the International Atomic Energy Agency (IAEA) in 1964 under an agreement with the Italian Government, and with the a ssis tance of the City and University of Trieste. The IAEA and the United Nations Educational, Scientific and Cultural Organi zation (UNESCO) subsequently agreed to operate the Centre jointly from 1 January 1970. Member States of both organizations participate in the work of the Centre, the main purpose of which is to foster, through training and research, the advancement of theoretical physics, with special regard to the needs of developing countries.
GLOBAL ANALYSIS AND ITS APPLICATIONS IAEA, VIENNA, 1974 STI/PUB/355
Printed by the IAEA in Austria September 1974 FOREWORD
The International Centre for Theoretical Physics has maintained an interdisciplinary character in its research and training, programmes in different branches of theoretical physics. In pursuance of this objective, the Centre has organized extended research courses with a comprehensive and synoptic coverage in varying disciplines. The first of these — on plasma physics — was held in 1964; the second, in 1965, was concerned with the physics of particles. Between then and 1972, seven courses were organized; three on nuclear theory (during 1966, 1969 and 1971), three on physics of condensed matter (during 1967, 1970 and 1972), and one on Computing as a Language of Physics (1971). The Proceedings of all these courses have been published by the International Atomic Energy Agency. The present three volumes record the Proceedings of the tenth course held from 4 July to 25 August 1972 which dealt with Global Analysis and its Applications. Generous grants from the United Nations Development Pro gramme and from the Battelle Foundation are gratefully acknowledged. The programme of lectures was organized by Professors M. Dolcher (Trieste, Italy), J. Eells (Warwick, United Kingdom) and J.C . Zeeman (Warwick, United Kingdom).
Abdus Salam EDITORIAL NOTE
The papers and discussions incorporated in the proceedings published by the International Atomic Energy Agency are edited by the Agency's edi torial staff to the extent considered necessary for the reader's assistance. The views expressed and the general style adopted remain, however, the responsibility of the named authors or participants. For the sake of speed of publication the present Proceedings have been printed by composition typing and photo-offset lithography. Within the lim i tations imposed by this method, every effort has been made to maintain a high editorial standard; in particular, the units and symbols employed are to the fullest practicable extent those standardized or recommended by the competent international scientific bodies The affiliations of authors are those given at the time of nomination. The use in these Proceedings of particular designations of countries or territories does not imply any judgement by the Agency as to the legal status of such countries or territories, of their authorities and institutions or of the delimitation of their boundaries. The mention of specific companies or of their products or brand-names does not imply any endorsement or recommendation on the part of the International Atomic Energy Agency. CONTENTS OF VOL. II
Geometric variational problems from a measure-theoretic point of view (IA E A -SM R -11/9) ...... 1 F .J . Almgren, J r . Area measures on a real vector space (IAEA-SMR-11 / 10) ...... 23 F . Brickell The differentiability of transformations which preserve geodesics (IA E A -SM R -11 / 11) ...... 33 F . Brickell Stability theorems for R2-actions on manifolds (IAEA-SMR-11/12) .. 37 C. Camacho Introduction to minimal-surface theory (IAEA-SMR-11/13) ...... 43 E . D e Giorgi On complex varieties of nilpotent Lie algebras (after G. Favre) (IA E A -SM R -11/14) ...... 47 P . D e l a Harpe On infinite-dimensional Lie groups acting on finite-dimensional m anifolds (IA E A -S M R -ll/1 5 ) ...... 59 P. D e l a Harpe Some properties of infinite-dimensional orthogonal groups (IA E A -SM R -11/16) ...... 71 P. D e l a Harpe T heory of re sid u e s in se v e ra l v a ria b le s (I A E A - S M R -ll/17) ...... 79 P . Dolbeault On connections (IA E A -SM R -11 / 18) ...... 97 P . Dolbeault Introduction to global calculus of variations (IAEA-SMR-11 / 19) ...... 113 H.I. E 1 i a s s о n Elementary survey of pseudo-differential operators and the wave-front set of a distribution (IAEA-SMR-11/20) ...... 137 R .J . Elliott Boundary value problems for non-linear partial differential equations (IA E A -SM R -11/21 ) ...... 145 R .J. Elliott Gaussian measures on Banach spaces and manifolds (IA E A -SM R -11/22) ...... 151 K.D . E 1 w o r t h у Sheaf cohomology, structures on manifolds and vanishing theory (IA E A -SM R -11/23) ...... 167 M .J. Field . Complex analysis ori Banach spaces (IAEA-SMR-ll/24) ...... 18Э M .J. Field Modern theory of billiards - an introduction (IAEA-SMR-11/25)...... 193 G. Gallavotti Some remarks on quasi-Abelian manifolds (IAEA-SMR-11/26) ...... 203 F . Gherardelli , A. Andreotti Life and death of the Bernstein problem (IAEA-SMR-11/27) ...... 207 E . G i u s t i Invariants of foliations (IAEA-SMR-11/28) ...... 215 C. Godbillon On the local solvability of linear partial differential equations (IA E A -SM R -11 /29) ...... 221 H. Goldschmidt Compact operators and the minimax principle (IAEA-SMR-11/30) ... 225 R.A . Goldstein, R. Saeks R igidity and energy (IA E A -SM R -11 / 31) ...... 233 R.A . Goldstein, P .J . Ryan Phase transitions in D-dimensional Ising lattices (IA E A -SM R -11/32) ...... 245 R.A . Goldstein, J . J . Kozak Differential calculus in locally convex spaces (IAEA-SMR-11/33) ... 263 R.A . Graff Singularities in "soap bubbles" and "soap films" (IA E A -SM R -11 / 34) ...... 271 Je an E . Taylor C au stics (IA EA -SM R -11/35) ...... 281 J . Guckenheimer The topological degree on Banach manifolds (IAEA-SMR-ll/36) ...... 291 C .A .S. Isnard Existence and non-existence for semi-linear elliptic equations (IA E A -SM R -11 /37) ...... 315 J .L . K a z d a n An example of a strange three-dimensional surface in C2 (IA E A -SM R -11 /38) ...... 323 J . J . Kohn A continuous change of topological type of Riemannian manifolds and its connection with the evolution of harmonic form s and spin stru c tu re s (IA E A -SM R -11/39) ...... 329 J . Komorowski A new proof for regularity of solutions of elliptic differential o p e rato rs (IA E A -SM R -11/40) ...... 355 M. Kuranishi S e c re ta ria t of Sem in ar ...... 363 lAEA-SMR-11/9
GEOMETRIC VARIATIONAL PROBLEMS FROM A MEASURE-THEORETIC POINT OF VIEW
F.J. ALMGREN, Jr. * Department of Mathematics, Princeton University, Princeton, N.J., United States of America
Abstract
GEOMETRIC VARIATIONAL PROBLEMS FROM A MEASURE-THEORETIC POINT OF VIEW. Some phenomena of geometric variational problems are treated; in particular, a discussion of surfaces as measures, a regularity theorem, and estimates on singular sets are presented.
"During the last three decades the subject of geometric measure theory has developed from a collection of isolated special results into a cohesive body of basic knowledge with an ample natural structure of its own, and with strong ties to many other parts of mathematics. These advances have given us deeper perception of the analytic and topological foundations of geometry, and have provided a new direction to the calculus of variations. Recently the methods of geometric measure theory have led to very substantial progress in the study of quite general elliptic variational problems, including the multidi mensional problem of least area. " [ 1 (Preface)].
This article, consisting of five parts, is intended as an introduction to the collection of mathematical techniques and results known as geometric measure theory from the point of view of certain problems arising in the calculus of variations.
PART A. SOME PHENOMENA OF GEOMETRIC VARIATIONAL PROBLEMS
Suppose k s n are positive integers and one is given a reasonable surface S of dimension к in Rn. Assume also one is given a suitable function
F : Rn X G(n, k) - R+
Here G(n, k) denotes the Grassmann manifold of all unoriented k-dimensional planes through the origin in Rn or, equivalently, the space of all k-dimensional
The preparation of this article was supported in part by a grant from the National Science Foundation and in part by funds from the Science Research Council in connection with the Symposium on Global Analysis, 1971 - 1972, at the University of Warwick.
1 2 ALMGREN tangent plane directions in Rn. Then one can define the integral F(S) of F over S by the formula
Z(S) = J F(p,Tank(S,p)) d g ? k p p £ S
Here Tank(S, p)e G(n, k) denotes the k-dimensional tangent plane direction to S at p for рб S, and.gfk denotes the k-dimensional Hausdorff measure over Rn defined in C. 3(1). The k-dimensional Hausdorff measure of a k-dimensional submanifold M of Rn of class 1 agrees with any other reason able definition of the к-area of M. However, with the use of the Hausdorff к-measure, one can make mathematically precise the notion of the k- dimensional area on surfaces which may have essential singularities.1 With this terminology, there are several problems one might wish to study. First problem (existence): Among all surfaces S having, say, a pre scribed boundary В (and possibly satisfying other constraints), is there one minimizing F(S)? Second problem (regularity): If there is a solution to the first problem, how nice is it? In particular, is it generally like a smooth k-dimensional submanifold of R11? Third problem (structure of singularities): What kind of singularities are possible in solutions to the first problem (if any)? And if singularities are possible, what language should one use to describe them? Fourth problem (computation): If there are solutions to the first problem, how does one explicitly find them? To make these problems mathematically precise, there are of course several questions to be settled, namely:
(1) What is a surface? (2) What is the boundary of a surface? (3) What are reasonable conditions to put on F? (We already have been assuming implicitly that F is measurable, for example. )
Before attempting to answer these questions, it is perhaps useful to consider the phenomena which arise when one studies these problems. In the following examples, the integrand F is restricted to being identically 1, i.e. for each (p, ж) e Rn X G(n, k), F(p, ir) = 1. The problem of minimizing F(S) thus becomes the problem of minimizing £ f k(S), in other words, the
1 There are many distinct k-dimensional measures over Rn. Although all of them are Borel regular, invariant under the Euclidean group, defined by reasonable geometric constructions, and agree with the Hausdorff measure on class 1 submanifolds, they can disagree completely on more general subsets of Rn. One reason for using the Hausdorff measure lies in the structure theorem for sets of finite Hausdorff measure [1(3.3.13)] which says that any Borel subset A of Rn for which ^ ( A ) <*> can be written almost uniquely as A = В U С, В П С = 0, such that В can be covered by a countable family of k-dimensional submanifolds of Rn of class 1 and S^ttC ) = 0 for almost all orthogonal projections эт: Rn -» R^; hereS^ denotes Lebesgue k-dimensional measure over R^, which equalsover R , This structure theorem, proved for к = 1, n = 2 by A.S. Besicovitch and in general dimensions by H. Federer, is one of the several main achievements of the geometric measure theory, and, typically, it plays an essential role in proving the existence of solutions to variational problems in the measure-theory context. IAEA-SMR-11/9 3
FIG.l. A disk with five handles.
problem of minimizing the к-dimensional area of S. The various problems associated with minimizing area are sometimes called collectively Plateau's Problem in honour of the Belgian physicist J. Plateau of the last century who, among other things, studied the geometry of soap films and soap bubbles (see [2 ]). Example 1. Suppose С is a fixed simple closed curve in R3 of finite length and let Do denote a fixed two-dimensional disk, say D0 = R2n {(x,y): x2 + y2 S 1}. The first person to make significant progress in a special formulation of the problem of least two-dimensional area was the late J. Douglas who showed in particular: Theorem [3]: Among all maps D0 ->■ R3 such that 3D0 maps homeo- morphically onto C, there exists a mapping of least two-dimensional area ("two-dimensional area" as used in this theorem is defined explicitly in B. 6). Unfortunately, there are no known direct extensions of this result to higher dimensional disks. Now for m = 1,2,3, ... let Dm denote a fixed two-dimensional disk with m handles (a D5 appears in Fig. 1), and let Am denote the infimum of the areas of the mappings Dm -» R3 such that 3Dm maps homeomorphically onto С (mappings realizing this infimum area do not seem to be known to exist). Since one can always "pinch out" a handle before mapping Dm, one has the obvious inequalities
A0 ê A, = A2 g ... й limm Am g 0.
For a curve like that sketched in Fig. 2 (a simple closed unknotted curve of finite length), W.H. 'Fleming has proved the strict inequalities
A0 > A, > A2 > ... > limm Am> 0 [4].
The significance of these inequalities is that if one wishes to solve the problem of minimizing area among all oriented surfaces having boundary C, then one cannot find an absolute minimum among oriented surfaces of finite topological type, since, indeed, the list of Dm's is a complete topological classification of all compact orientable two-dimensional manifolds having a circle as boundary. On the other hand, there is a surface S (sketched in Fig. 3) which perhaps deserves to be called the oriented surface of least 4 ALMGREN
FIG.3. The oriented surface S having boundary С and being of least area is of infinite topological type.
area having С as boundary. S ~ С is a two-dimensional real analytic submanifold of R3 having 0 mean curvature at each point and the area of S equals lim m Am (see E. 1). The surface S is of infinite topological type and is homeomorphic with the surface T sketched in Fig. 4. (The nature of this surface T suggests why, frequently, homological conditions in geometric measure theory are stated in term s of the Vietoris or Cech theories rather than the singular theory). Example 2. Suppose В consists of three points in R2 which are the vertices of an equilateral triangle with centre at 0 e R2 . Then among all one-dimensional sets S which are the unions of non-trivial rectifiable arcs and through which each point of В is pathwise connected to each other point, the unique set Y (see Fig. 5) of least total length consists of the union of the three line segments connecting the points of В to 0 (the proof of this is not completely trivial). This problem is as naturally posed as one could ask, the integral (length) is real analytic, the boundary В is algebraic, and the solution Y is unique. Nevertheless, the solution has an interior singularity of codimension 1, namely 0 e Y с R2 where the three line segments meet. Example 3. Suppose V is a complex algebraic variety of complex dimension к in complex n-dimensional space (En, and let U c Œn be open and bounded such that 9(V nU) is suitable (in the language of C.3(4), VnU is required to be a 2k-dimensional integral current in R = Œn); the complex structure of V, of course, gives V a natural orientation so that V nU together with that orientation becomes an oriented 2k) rectifiable and ^ ^ k-measurable subset of R2n [ 1(4. 2.29)]. Let W be any other oriented IAEA-SMR-11/9 5
FIG.4. A disk T with infinitely ma n y handles converging to a boundary point.
FIG. 5. The 1-dimensional set Y connecting the three points of the boundary В and of least length has an interior singularity of codimension 1.
2k-dimensional surface (integral current) such that 3W = 9(V n U). Then either W = V n U or the 2k-area of W (M(W)) is strictly larger than the 2k-area of V (lU (M(V n U)). Furthermore, this is true whether or not V has singularities. In particular, if one wishes to solve the problem of minimizing oriented area, and really achieve the least area, then one must at times admit ás singularities in the solutions to Plateau's problem at least all the singularities occuring in complex algebraic varieties. Example 4. R. Thom has constructed a 14-dimensional compact real analytic manifold M without boundary with a seven-dimensional integral homology class и which cannot be represented by any seven-dimensional smooth submanifold [ 5] . On the other hand every integral homology class of any compact smooth Riemannian manifold can be represented by an oriented surface (integral current) of least area (mass) in that class. Thom's example thus shows that sometimes there can be topological obstructions to surfaces of least area being free of singularities. Example 5. Consider the following partitioning problem. Let т х, тг, ... , т 4> 0. Then among all disjointed regions A1; ... , A ,cR “ such that ¿^(Aj) = m¡ for each i, are there regions for which
9 A i) i = 1 attains a minimum value? This problem always admits solutions [6] and general arguments (see D.3) show that except for a (possibly empty) compact 6 ALMGREN
FIG. 6. The unique minimal partitioning configuration for two regions of prescribed volumes.
FIG. 7. A "soap-bubble-like” minimal partitioning configuration consisting of six real analytic surfaces meeting at 120° along four smooth arcs which meet at equal angles at two points. singular set of zero ^ n-1 measure, U^Aj is a Holder continuously differ- entiable n-1 dimensional submanifold of Rn. The particular form of the problem above further implies the real analyticity of the regular part of Ui8Ai. For i = 1, the unique solution to this problem (up to isom etries of Rn, of course) is a standard n-ball of the prescribed volume. For i = 2, n = 3, the unique solution (sketched in Fig. 6, also sketched "blown apart") consists of three spherical pieces meeting along a circle (in this case the circle is the compact singular set of zero Çf2 measure referred to above and in D. 3). For t = 3, n = 3, the solution seems to be that sketched in Fig. 7; note that six pieces of real analytic surface meet tangentially at 120° along four smooth arcs which in turn meet at two vertices tangentially as the central cone over the vertices of a regular tetrahedron (see [ 2] ). IAEA-SMR-11/9 7
FIG. 8. The Môbius band as a surface of least area.
FIG.9. The triple Mflbius band as a surface of least area.
Example 6. A Mobius band-like surface (sketched in Fig. 8) occurs as a soap film for a wire bent in the shape of the boundary shown, and also occurs as a surface of least area among all mathematical surfaces spanning sue!} a boundary in the sense of homology with coefficients in the integers modulo 2 (see E.2). Similarly a triple Mobius band-like surface (sketched in Fig. 9) occurs as a soap film for a wire bent in the shape of the boundary shown, and also occurs as a surface of least area among all mathematical surfaces which span such a boundary in the sense of homology with coefficients in the integers modulo 3 (see E.3). Finally a surface S like that sketched in Fig. 10 (like a Mobius band on the left joined to a triple Mobius band on the right by a thin ribbon of surface, having as boundary С a single simple closed unknotted curve) occurs both as a soap film and as a mathematical minimal surface. However, J.F. Adams has pointed out the existence of a continuous retraction (S, C) -* (С, C) of S onto the boundary С [ 7 (Appendix)] so that, in no way, in the sense of algebraic topology, does S "span" C. In particular, if one wishes to regard 8 ALMGREN
FIG. 11. A soap film with a boundary wire which is not closed.
C as the boundary of S, then one must use other definitions of boundary of a surface than those of algebraic topology (see, in particular, the variational formulation in [ 8] ). Example 7. Suppose one bends a wire into the shape of an overhand knot as sketched in Fig. 11(a) (note that the two ends of the wire are free). Typically, when such a wire is dipped in soapy water, a film such as that IAEA-SMR-11/9 9 sketched in Fig. 11(b) forms, even though the wire is not closed. Such a film does admit a mathematical approximation, but only with a "boundary" of substantial positive thickness. Indeed one can prove by tangent cone arguments (see the nice discussion of such cones in Ref. [ 2] ) that such a mathematical surface is impossible over an infinitely thin boundary of class 3. The significance of this, among other things, is that if one wishes to construct a theory of minimal surfaces which in particular includes the phenomena suggested by soap films, then one must at times admit boundaries of sub stantial positive thickness. Example 8. Suppose В = {(-2, 0), (2, 0)} cR 2. Then among all one dimensional sets S lying in R2 n {(x, y): x2 + y2 S 1} , which are unions of non-trivial rectifiable arcs through which the points of В are pathwise connected to each other, there are exactly two distinct sets С and С' of least total length, where C-^ = L1UL2UL3,
L x = {(x, у): у = 3'* (2 + x) , -2 S x á -3^}
4 = {(x,y): у = 3_i (2 - x) , 31 i x é 2}
L3 = {(x, у): у = (1 - x2)", -3‘* â x â 3^} and С1 is the image of С under reflection across the x-axis (see Fig. 12). This problem is naturally posed, the integral (length) is real analytic, the boundary В and the obstacle {(x, y): x2 + y2 < 1} are algebraic, and there are exactly two solutions (a trivial modification makes the solution unique). Nevertheless each solution curve, although a one-dimensional submanifold of R2 of class 1, is not a submanifold of class 2. The tangent lines of С and C, however, do vary in a Lipschitzian manner, hence a fortiori Holder continuously (see D.3). Example 9. Suppose В = {(0,-1), (0, 1)}. Then the unique one dimensional set С in R2 of least length connecting the points of В is, of course, С = R2 n {(x, у): x = 0, -1 s y û 1} (see Fig. 13(a)). Now let f: R2 -► R2 be the algebraic diffeomorphism given by f(x, y) = (x + y 3, y).
FIG. 12. The curve С of least length is not of class 2. 10 ALMGREN
The length integrand F: R2 X G(n, 1) -* {1} transform s naturally under f to give a new integrand G = f#F with the obvious property that the unique one dimensional set D of least G integral among those sets connecting the points of f(B) is D = f(C). Note that D is also the graph of the function y = for -1 û x § 1 (see Fig. 13(b)). The ellipticity (D. 1(1)(2)) of F implies the ellipticity of G, and it is clear that the function y = x^ is the unique natural solution to the real analytic elliptic Euler-Lagrange differential equation associated with G and the standard (x, y) coordinates for R2 (see D.2). This function is not even of class 1, however (although it is real analytic except for a compact singular set of zero measure — a representative conclusion for such problems (see C. l(5)(d))).
PART B. GEOMETRIC VARIATIONAL PROBLEMS IN A MAPPING SETTING AND ASSOCIATED VARIFOLDS
В. 1. Variational problems in a mapping setting
Suppose one is given a suitable open set W in Rk. We will denote by „
Ф: -► R. In case к = 0, n = 1, then Л can be regarded as, say, the space of class 1 mappings Ф: R -» R, and the basic problem which led to the differential calculus was that of finding a point where Ф assumes its maxi mum value. Equivalently one could seek those points where Ф takes its minimum value, or, more generally, one could seek critical points of Ф, i. e. points at which the first derivative Ф' vanishes. In case k â 1, n i 1, then, heuristically at least, the basic problem of the calculus of variations (in this context) is that of finding points (actually mappings f: closure W -> Rn) at which (most commonly) Ф assumes its minimum value, or, more generally, the critical points of Ф, i.e. where the first variation 6Ф vanishes identically (in practice the definition of first variation varies considerably from problem to problem). In a number of ways, as the above phraseology suggests, there are analogies between calculus and the calculus of variations. The ordinary calculus has been extended to differential manifolds in various fashions, and it is sometimes useful to regard certain problems in the calculus of variations in the language of the ordinary calculus, but extended to manifolds having infinite dimensions. These manifolds of infinite dimension typically are "modelled on" Hilbert spaces or Banach spaces. Unfortunately, such infinite dimensional manifolds so far have not played a significant role in the geometric variational problems with which we are mainly concerned here.
B. 2. Special forms of the function Ф
The functions Ф;„<^ -► R which have received the greatest mathematical attention in higher dimensions are integrals of the form
(*) Ф(Г) = J H ere к к Another example is the k-dimensional area integrand cpA which is defined by settin g cpA(x, f(x), Df(x)) = II Лк Df(x) I Here AkDf(x): AkRk -> AkRn and ||AkDf(x)|| is equal to the square root of the sum of the squares of all the к by к minors of the n by n Jacobian matrix 12 ALMGREN 9f1 9f1 df} (x) (x) (x) 9x- 9 x 2 Эх, 9 f2 , . 9 f 2 9f 9 Z (X) 9x0 (X) Эх., (X) 9fn 9 fn 9f (X) (x) Эх-, (x) 3xn 9xi As indicated, we are prim arily concerned with geometric integrals Ф corresponding to geometric integrands q>. Several definitions are in order. B.3. Definitions (1) A function Ф: Л -+ R+ is called an integral in parametric form if and only if Ф has the form (*) above and, in addition, for each fç.jX and each diffeomorphism g: W -» W, 0(f) = S(fog) (2) A function F: Rn X G(n, k) - R+ such that for each f e. $(f) = J F[p, Tank(f(W), p)] N(f,p) d ^rkp p e f ( W ) Here N(f, p) = card f 1{p} for each pef(W ). (4) A function ip: W X Rn X Hom(Rk, Rn ) -» R is called a geometric integrand if and only if the function Ф defined by (*) is a geometric integral. В. 4. Remark To the best of my knowledge the expression "integral in parametric form" was introduced by C.B. Morrey, Jr. to suggest that the appropriate integrals to be studied over parametric surfaces (surfaces which are not IAEA-SMR-11/9 13 the graphs of functions) are integrals in parametric form [9]. This term i nology has been a source of confusion, however, since, in particular, integrals in param etric form are precisely the integrals of form (*) which do not depend on the param etrization of the domain. For sufficiently nice f it is clear that any geometric integral has the invariance property characteristic of integrals in parametric form. In case the maps in Л are Lipschitzian and Tank(f(W),p) is understood to m ean the (,g^kL f(W ), k) ap p ro x im ate tang ent cone, T an k( ^ kL_ f(W), k) [1(3.2.16)], then any geometric integral is indeed an integral in parametric form. Also, in case Ф is an integral in parametric form which is of class 2, range $cR +, and Ф is independent of orientation, thenC.B. Morrey, Jr. has shown that $ is a geometric integral [ 9]. Integrals in parametric form, and geometric integrals in particular, are of special geometric significance since the value of the integral depends only on the geometry of f(W) and not on the particular parametrization which produces it. I know of no natural geometric, physical, or biological problems involving integrals in param etric form which are not also geometric integrals. The к-dimensional area integrand is an example of a geometric integrand while the Dirichlet integrand is neither a geometric integrand nor an integrand in parametric form. One refers to a variational problem as being a geometric variational problem in case the only integrals which arise in the problem are geometric in te g ra ls . B. 5. Some difficulties with geometric variational problems The greatest initial difficulty with the study of geometric variational problems (in the mapping setting) is that, in general, there is no straight forward way to obtain solutions by the so called direct method of the calculus of variations. The term "direct method of the calculus of variations" is used loosely in a number of different contexts. Roughly speaking the "direct method" makes sense in a variational problem when the formulation of the problem itself guarantees the existence of a sequence {fjlj of mappings in„^, any suitable convergent (in„ B.6. Examples Suppose к = 2, n = 3, W R 2 n {(x, y): x 2 + у2 < 1}, consists of all Lipschitzian mappings f: closure W R3 such that f(x, y) = (x,y,0)ERS w henever x2 + у 2 = 1, and 2 "af1af2 af1 af2" "af1 9f 3 dî1 9f3]2 ~8f2 af3 af2af5 i( f ) + dx dy _Эх Эу Эу Эх. . Эх Эу эу 9xJ + _Эх Эу Эу 9x_ 14 ALMGREN for fe„/f. Ф is, of course, the two-dimensional area integral. Suppose one is given the geometric variational problem of finding he..^such that Ф(Ь) = inf Ф. The mapping h(x,y) = (x, y, 0) is, of course, such a mapping. However, in general, there is no reason for a minimizing sequence to admit a reasonable convergent subsequence as the following examples show: (1) Let pj , p2, p3, ... G R3 be a list of the points in R3 with rational coordinates. One can construct a minimizing sequence {f±}¡ with linij ®(fj) = in f Ф = ir such that {px, p2, . . . , p¡} с f¡(W) for each i = 1, 2, 3, for example by deforming the map h above by pulling thin "tentacles" out of the flat disk h(W) in R3 as indicated in Fig. 14, the total area of which "tentacles" is no more than 1/i. In this case, each point in R3 is a limit point of the sequence {f^W)}^ (2) Let W be given the usual polar coordinates (r, 6) and define for i = 1, 2, 3, ... the diffeomorphisms g : W W gj(r, 0) = ((1 /i)r + (1 - l/i)r‘, в) Clearly, the sequence {hogj}¡ is a minimizing sequence for the above problem. However, lim^hogj) (x, y) = (0,0,0)£R3 for x2 + y2 < 1 limithogj) (x, y) = (x,y, 0) for x2 + y2 = 1 so that no suitable convergent subsequences exist. FIG 14. A disk with tentacles. IAEA-SMR-11/9 15 В. 7. Geometric integrals from another point of view Suppose Ф: -* R+ is a geometric integral, F: Rn X G(n, k) -» R+, and for each f e ®(f) = J Ftp, Tank(f(W),p)] N(f,p) d ^ p p e f(W) Now define f o r ^ kl_f(W) almost all p e Rn, ф{(p) = (p, Tank(f(W), p))e Rn X G(n, k) and note that one can write ®(f) = f Fo 3>(f) = / F d {ф{# [ ^ kLN(f,.)ll Rn x G(n, k) in w hich ф ^ [gf LN(f, . )] determines a unique Radon measure over Rn X G(n, k) (assuming, say, W is bounded and f is Lipschitzian). Note that this measure is independent of Ф and F. A definition is in order. B. 8. Definition [ 8] By a k-dimensional varifold in Rn one means a Radon measure over Rn X G(n,'k), Vk(Rn) denotes the space of all k-dimensional varifolds in Rn. In case W is a bounded open set in Rk and f: W -» Rn is Lipschitzian, one den otes by f#|w |eV k(Rn) the varifold corresponding to ¡//f# [^"kL N(f, . )] as in B. 7. above. B.9. Remark As was noted in B. 7, ®(f) = J f d f#|w| . One can also verify, in a reasonably straightforward manner, that for each and each diffeomorphism g: W -*• W, f # |w | = (f°g)# IW 16 ALMGREN so that clearly the mapping is not uniquely determined by its associ ated varifold |w| . Of interest, however, is the fact that spaces of Radon measures have very strong convergence properties in the weak topology. In the present context, we have: B. 10. Definition A sequence {Vjj of Radon measures over Rn X G(n, k) (i.e. a sequence of elements of (Rn)) is said to converge weakly to a limit Radon measure V if and only if for each continuous function F: Rn X G(n, k) -* R* with compact su p p o rt, lim i / F dV¡ = / F dV B. 11. Proposition A sufficient condition that a sequence {V1}i of Radon measures over Rn X G(n, k) contains a subsequence {V[.}. which converges weakly to some lim it Radon measure V is that for eachJ 0 < r < oo, sup j Vi [Bn(0, r) X G(n, k)] < oo В .12. Examples Let us return to the examples of B.6 above which illustrated the diffi culties of the attempt to use the direct method in geometric variational problems. We have corresponding examples. (1) Let {fJi be as in B. 6(1). Then it is easily verified that lim ¡ f i# IW I = h#|w| (weakly) where h(x, y ) = (x, у, 0) as in В. 6, is a solution to the problem in В. 6. Geometrically, h#[w| corresponds to the unit flat disk in R2 X {0}cR3 which "solves" the problem, but without a particular parametrization prescribed for that disk. (2) Let {g.}. be as in B. 6(2) above. Then also limj (hog¡)# | W| =h#|w| (w eakly). Indeed, for each i, (hogj)# |w| = h#|w| . PART С. SURFACES AS MEASURES С. 1. Reasons for the measure theoretic approach As has been suggested by the various examples and discussion in Parts A and B, one approach to the study of geometric variational problems is based on a correspondence between suitable surfaces and measures on appropriate spaces. Indeed, the natural setting for geometric problems IAEA-SMR-11/9 17 in the calculus of variations seems to be that in which surfaces are regarded as intrinsically part of Rn (in particular as measures on spaces associated with Rn) rather than as mappings from a fixed k-dimensional manifold, even though with this approach one is not able to use the traditional methods of functional analysis for showing the existence of solutions. The main reasons for doing this are the following: (1) Mappings from a fixed compact k-dimensional manifold cannot take into account the phenomena of the examples given in Part A. For example, one cannot consider surfaces of infinite topological type, surfaces having singularities not realizable by mappings like the singular curve of the triple Mobius band, or surfaces having boundaries defined in certain w a y s. (2) Many significant results have been obtained from the study of geometric variational problems in the measure theoretic setting in contrast with the virtual absence of such results in higher dimensions and codi mensions in the mapping setting [1, 2, 6, 8, 10]. (3) It seems reasonable to hope that once the singularities of the measure theoretic solutions are understood, one will be able to solve the mapping problem as a consequence. (4) Topological methods analogous to those of M orse's theory are available in the measure theoretic setting in contrast to the absence of such methods in the mapping setting [ 11]. For example, the "Condition C" of Palais and Smale [12] is not satisfied by any geometric variational problem (note B.6). (5) The techniques developed in the study of geometric problems in a m easure-theoretic setting have aided in the solution of related problems. For example: (a) The basic theorems of classical integral geometry have been proved in what seems to be their most natural setting [ 13]. (b) Various long standing questions in the theory of Lebesgue area have been settled [ 14]. (c) Extensions of Bernstein's theorem that a globally defined non- parametric minimal hypersurface must be a hyperplane have been proved in dimensions up to 8 and counterexamples have been shown to exist in higher dimensions [ 1 (5. 4. 18), 15] . (d) Proofs have been given of the regularity almost everywhere of "weak" solutions to some non-linear elliptic systems of partial differential equations, and examples exhibited which show some times unique solutions to such systems which contain essential discontinuities [16, 17]. (6) In the measure-theoretic setting, many important natural geometric constructions are available, the analogues of which in spaces of mappings seem unnatural. C.2. Remark The geometric measure theorist routinely works with a number of different measures and measure theoretic surfaces. The following defi nitions and terminology are both representative of the most common "surfaces" of the geometric measure theorist and enable a careful statement to be made of some of the regularity and singularity results in Parts D and E. 18 ALMGREN C.3. Definitions and terminology Suppose 0 s k è n are integers. (1) For each A с Rn( the k-dimensional Hausdorff measure of A, denoted is the greatest lower bound of all t such that 0 s t s oo and for every e > 0 there exists a countable covering G of A with s s G and diam (S) < e f o r S s G ; h e re a(k) = i / k(Rk n {x: | x | < 1}) [ 1 (2)]. (2) A subset AcRn is called { ^ k, k) rectifiable if and only if ^И'(А) < oo and there exists for each e > 0 a compact k-dimensional differential sub manifold M£ of Rn of class 1 such that ~ M£ ) < e. In case A is £fk measurable one can choose Me so that ■$fk([A ~ Me] и [Me ~ A]) < e. (3) In c a se A с Rn is (gfk, k) rectifiable and measurable, II Aj| = ^ KLA denotes the measure over Rn defined by requiring ( ^ k L A) (B) = n B) for each В с Rn, and Tank(.grkLA, a) denotes the (£?k, k) approximate tangent cone to A at a. [ 1 (3. 2.16)]. For k L_ A almost all a in Rn, Тапк(^И*1_ A, a)e G (n, k). An orientation for A is an gfkLA measurable function u: A -» G0(n, k) such that for !gfk alm o st a ll a e A, you (a) = Tank0g^kLA, a). Here G0(n, k) is the Grassmann manifold of all oriented k-dimensional planes through the origin in Rn and y: G0(n, k)-*G(n, k) is the natural projection. The pair (A, со) is called an oriented (gfk, k) rectifiable and measurable subset of Rn. (4) If A cRn is (gfk, k) rectifiable and $fk measurable, the integral varifold associated with A, denoted | A |, is the Radon measure over Rn defined by [lRn X Tank(^ kLA, ,)]# [^ kLA] A Radon measure V eV k (Rn) is called an integral varifold if and only if it is a finite or convergent infinite sum of measures {|Aj|}. corresponding to (gfk, k) rectifiable and measurable subsets Ах, A2, A3, ... of Rn. The most accessible source for the basic theorems about varifolds is Ref. [ 8] to which the readers attention is strongly directed. (5) A k-dimensional current in Rn is a continuous linear functional on the space p e a Here we are identifying G0 (n, k) with the submanifold of all simple unit к-vectors in the Grassmann vector space AkRn of all к-vectors in Rn. IAEA-SMR-11/9 19 A k-dimensional current T in Rn is called rectifiable if and only if it is the finite or convergent infinite sum of rectifiable currents associated with oriented (^ k, k) rectifiable and measurable subsets of Rn as above. A k-dimensional current T is called a k-dimensional integral current in Rn if and only if both T and ЭТ are rectifiable currents (for к = 0, there is no requirement regarding ЭТ which, of course, is not defined). The most basic theorems about integral currents in general are the deformation theorem [1 (4.2.9)], the compactness theorem [1 (4.2.17)], and the approximation theorem [ 1 (4.2.20)]. The reader's attention is strongly directed to these fundamental facts. The compactness theorem alone is the essential ingredient in many existence results for elliptic geometric problems by the direct method. The necessary lower semi continuity follows from the ellipticity. One usually takes as the "k- dimensional area" of a k-dimensional integral current T in Rn the mass of T, denoted M(T), defined by setting M (T) = sup {T( Note that M(2T) = 2M(T) so that, in particular, the mass of T can be much larger than ^fk(spt T). (6) In a natural way rectifiable currents may be regarded as having the integers as "coefficient group". For each integer v г 2, the k- dimensional flat chains modulo v in Rn are defined essentially as the quotient subgroup of the group of k-dimensional rectifiable currents in Rn induced by the homomorphism Ж -* Ж ¡(2 Ж). Theorems corresponding to the defor mation theorem, the compactness theorem, and the approximation theorem hold for flat chains modulo v [1 (4. 2. 26)] . PART D. A REGULARITY THEOREM In this part we give a careful statement of a very general regularity result for solutions to elliptic geometric variational problems. D. 1. Definitions (1) An integrand F : G(n, к) -» R+ is called elliptic with ellipticity bound с if and only if с > 0 and the following condition holds. Suppose D с Rn is a flat k-dimensional disk and S is a compact \ k) rectifiable subset of Rn which can not be mapped into 3D by any Lipschitzian function f: Rn -* Rn such that f(p) = p for peSD (it follows that 9DcS). Then F(S) - F(D) g с LgÉ^ÍS) - (D)] (2) An integrand F: Rn X G(n, k) -* R+ is called elliptic if and only if there is a continuous function c: Rn -> R+ such that for each p eR N, Fp is elliptic with ellipticity bound c(p); here Fp: G(n, k) -» R+, Fp(7r) = F(p, ir) for TreG(n, k). 20 ALMGREN (3) Suppose 0 Sa < 1 and ej R+ -*■ R+ is monotonically non-decreasing w ith 1 J t ‘(1 + 0° ejt)1 dt < oo о Suppose also F: Rn X G (n, k) -<■ R+ is elliptic and of class 3. Let ScR n be k) rectifiable and BcRn ~ S be closed. One says that S is (F,eJ minimal with respect to В if and only if there exists 6 > 0 such that F (S n { z : (a) tp: Rn -> Rn is Lipschitzian, (b) [{ z: cp{z) f z} u cp {z: D.2. Remark In case к = n - 1 the ellipticity of F in D. 1(2) above is equivalent to the uniform convexity of F p for each p e Rn. For all k, the set of elliptic integrands F as in D. 1(1) contains a convex neighbourhood in the class 2 topology of the к-dimensional area integrand. Also for each diffeomorphism f: Rn ->• Rn , f#F is elliptic if and only if F is, for f as in D. 1(2). Finally the ellipticity of F implies that the various Euler-Lagrange equations which arise are non-linear strongly elliptic systems of partial differential equations, and, "in the small", the ellipticity of F is equivalent to the strong ellipticity of these equations. D. 3. Theorem [ 6] S uppose a , F, S, В are as in D. 1(3) above and that S is (F, e.) minimal with respect to B. Then: (1) ^ k([S ~ spt(^kLS)] и [sptigH'LS) ~ (SuB)])= 0 (2) There exists an open subset U of Rn such that: (a) ^ k[spt(^kL.S) - (UuB)] = 0, (b) sp t(^ kl_S)nU is a к-dimensional submanifold of Rn of class 1 with tangent к-planes which locally vary Holder continuously with exponent a . The special interest of this theorem is that the hypotheses are sufficiently weak to apply to all the natural constrained geometric variational problems of which I know, the solutions of which yield rectifiable sets. The surfaces discussed in examples 2, 5, 6, 7, 8 in Part A come into this category in particular. IAEA-SMR-11/9 21 PART E. ESTIMATES ON SINGULAR SETS Very little is known at the present time about the structure of the singular sets of solutions to general elliptic geometric variational problems, except for their existence. However, for the area integrand there has been substantial progress. The following three theorems are representative of the present state of knowledge. E .l. Theorem [10] Let В be an n-2 dimensional integral current in Rn such that ЭВ = 0. Then: (1) There exists an n-1 dimensional integral current T in Rn such that ЭТ = В and M(T) = inf {M(S): S is an n-1 dimensional integral current in Rn and 3S = B}. (2) There exists an open set U in Rn such that: (a) The Hausdorff dimension of spt T ~ (U и spt B) is at most n-8. (b) spt TnU is an n-1 dimensional real analytic submanifold of Rn having 0 mean curvature at each point. E. 2. Theorem [ 10] L e t lskln and let В be a k-1 dimensional flat chain modulo 2 in Rn such that ЭВ = 0. Then: (1) There exists a k-dimensional flat chain T modulo 2 in Rn such that ЭТ = В and M(T) = inf{M(S): S is a k-dimensional flat chain modulo 2 in Rn and 9S = B}. (2) There exists an open set U in Rn such that: (a) The Hausdorff dimension of spt T ~ (U.u spt B) is at most k-2. (b) spt TnU is a k-dimensional real analytic submanifold of Rn having 0 mean curvature at each point. E.3. Theorem [18] Let В be a 1-dimensional flat chain modulo 3 in R3 such that ЭВ = 0. Then: (1) There exists a 2-dimensional flat chain T modulo 3 such that ЭТ = В and M(T) = inf {M(S): S is a 2-dimensional flat chain modulo 3 in R3 and dS = B}. (2) spt T ~ spt В = AUC where: (a) A n С = 0. (b) С is a 1-dimensional submanifold of R3 of class 1 with tangent lines which locally vary Holder continuously. (c) A is a 2-dimensional real analytic submanifold of R3 having 0 mean curvature at each point. (d) Each point p in С has an open neighbourhood ^su ch that ^/nA has exactly three components, say Aj, A2, A3. Furtherm ore for each i, A[U(yrnc) is a Holder continuously differentiable manifold with boundary, and whenever i f j. A¡ m e e ts Aj along Ж п С at an angle of 120°. 22 ALMGREN E. 4. Remark See Ref. [2] for a complete classification of the interior local structure of mathematical "soap bubble" and "soap film"-like surfaces, including the first mathematical verification of the century-old "axioms of Plateau". REFERENCES [ 1] FEDERER, H ., Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Band 153, Springer-Verlag Berlin - Heidelberg - New York (1969). [2] TAYLOR, Jean E., Singularities in "soap bubbles" and’’soap films”, these Proceedings. [3] DOUGLAS, J., Solution of the problem of Plateau, Trans. Amer. Math. Soc. 33 (1931) 263 -321. [4 ] FLEMING, W .H ., An exam ple in the problem of least area, Proc. Amer. M ath. Soc. 7 (1956) 1065-1074. [5] THOM, R., Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954) 17 -8 6 . [ 6] ALMGREN, F. J ., Jr., Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, preprint, 294 pages. [7] RE1FENBERG, E.R., Solution of the Plateau problem for m-dimensional surfaces of varying topological type (with Appendix by J.F. Adams), Acta Math. 104 (1960) 1 -92. [8] ALLARD, W .K., On the first variation of a varifold, Ann. of Math. 95 (1972) 417. [9] MORREY, C.B., Jr., Multiple Integrals in the Calculus of Variations, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Band 130, Springer-Verlag Berlin — Heidelberg — New York (1966). [ 10] FEDERER, H ., The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension, Bull. Amer. Math. Soc. 79 (1960) 761 -771. [ 11] ALMGREN, F. J., Jr., The theory of varifolds. A variational calculus in the large for the k-dimensional area integrand (multilithed notes), Princeton (1964). [12] PALAIS, R .S ., SMALE, S ., A generalized Morse theory, Bull. Amer. M ath. Soc. 70 (1964) 165-172. [13] BROTHERS, J.E ., Integral geometry in homogeneous spaces, Trans. Amer. Math. Soc. 124(1966)480-517. [14] FEDERER, H ., Currents and area, Trans. Amer. Math. Soc. 98 (1961) 204-233. [ 15] BOMBIER1, E ., DE GIORGI, E ., GIUSTI, E ., M inim al cones and the Bernstein problem, Invent. M ath. 7 (1969) 24 3 -2 6 8 . [16] MORREY, C.B., Jr., Partial regularity results for non-linear elliptic systems, J. Math. Mech. 17 (1968) 649 - 670. [17] GIUSTI, E ., MIRANDA, M ., Sulla regolariti delle soluzioni deboli di una classe di sistemi ellittici quasi-lineari, Arch. Rat. Mech. Anal. 31 (1968/69) 173 -184. [ 18] TAYLOR, J ., Regularity of the singular sets of 2-dimensional area minimizing flat chains modulo 3 inR3, Invent. Math. 22(1943) 119. IAEA-SMR-11/10 AREA MEASURES ON A REAL VECTOR SPACE F. BRICKELL Institute of Mathematics, University of Southampton, United Kingdom Abstract AREA MEASURES ON A REAL VECTOR SPACE. In this paper, area measures, transversality and angular metric are treated. After a discussion of theorems applying to one-dimensional measures, there follow sections on scalar products associated with regular area measures as well as on area measures on manifolds. INTRODUCTION In this paper, we discuss some problems in differential geometry which arise in the theory of m-dimensional area on a real vector space of dimen sion n (1 Sm 1. AREA MEASURES, TRANSVERSALITY AND ANGULAR METRIC Let V be a real vector space of dimension n with scalar multiplication written on the right. An m-frame in V is an ordered set of m linearly independent vectors e ^ e j , , em]. The set of all such m-frames can be made into a differentiable manifold, the Stiefel manifold of m -fram es in V. We denote it by Vm n. The group L^ of non-singular mxm matrices with positive determinants acts on Vm n on the right according to the rule e = [e 1 ; ------e m] - [ e 1, ------i m] = еф w here e 8 = У| еа^В a and ф is the matrix [0“], The quotient space is the Grassmann manifold of oriented m-planes in V. We denote it by Gmin. 0 23 24 BRICKELL An area measure of dimension m on V is a positive differentiable function L on Vm n which satisfies the homogeneity condition, Ы е Ф) = (d et(//) L(e), ф £ Ь +т ( 1. 1) Example. An area measure of dimension 1 is a positive homogeneous function L of degree 1 defined on Vj n = V - 0. Example. A scalar product on V defines an area measure of dimension m for each m, 1 Sm and the functions pj, determine a global chart on Vm n. We write ff=3f/3p^ for any real valued differentiable function f on Vm n . With this notation it can be shown that the homogeneity condition (1.1) implies that ( 1. 2) w here 6g = 0 (af¡3), б “ = 1. Consider a plane 3rGGm n and choose an m-frame e in ir. A vector J e .V is transversal to w if 2 L “(e) X1 = 0, a = 1, . . . , m. It can be i i shown that this definition is independent of the choice of e in n and of the basis [Ej, . . . , E n]. The set of all vectors transversal to v fo rm an (n-m)-plane (non-oriented) called the transversal plane to v. The angular metric is a quadratic differential form on Gm n which we define in the following way. Put . . , m and consider the quadratic differential form on Vm n It can be shown that H is independent of the basis [E1, . . . , En]. Further, if ш: Vm n G mjI1 is the natural projection then H is the image under ¡3* of a quadratic differential form on Gm_n. We call this form the angular metric. IAEA-SMR-11/10 25 Example. Suppose that m = 1, n = 2 and introduce polar co-ordinates r, ti on Vi 2 =V - 0. We also regard в as a local co-ordinate on Gli2, the mani fold of oriented 1-planes or directions in V. The area measure L can be expressed as L =rf(0) where f is a positive differentiable function of Э, and a calculation shows that the angular m etric is (l + (f" / f ) ) d 6*2 It is natural to ask the extent to which transversality or the angular metric determines the area measure. It is very easy to prove by local arguments that L and L determine the same transversal planes if, and only if, L is a constant multiple of L. On the other hand, there are values of m ,n for which the proof of the following theorem needs a global argu ment [1-3]. Theorem 1. Two area measures, L, L determine the same angular metric if, and only if, L is a constant multiple of L. Proof. The sufficiency of the condition is obvious. We will prove its necessity for area measure of dimension 1 (the method extends to the general case). We define a function a on V - 0 by L = aL. The homogeneity condition on L, L implies that a =ct ° w where a is a differentiable function on the compact space Gj n. Consequently a attains a maximum value in V - 0. On the other hand, since L, L determine the same angular metric, a satisfies the partial differential equations Э2а 1 ida 9L 9a 3L i . . 9p¡ 3pJ Ь'Эр‘ 9рГ 9pi 9pï'“ J l,J ------П where we have written p1 for . It follows that a satisfies the elliptic equation V 92 q 2 V 9j 9L L (Эр1)2 L L Эр1 9pi i i and therefore the maximum principle of Hopf [4, p. 61] implies that a is a constant function. We remark that the previous example shows that a global argument is certainly necessary in the case m = 1, n = 2. For if f is any linear function then the locally defined measure L = rf(0) has the same angular metric as L = r. An area measure is said tobe regular if its angular m etric is positive definite. An equivalent condition is that H be positive semi-definite of rank m(n-m). In the case m = 1 a symmetric regular area measure is a norm on V. From now on, we shall be concerned only with regular area measures. Another natural question of a general nature is whether there is a duality between area measures of dimensions m and n-m. To answer this we intro duce the idea of a conformai area measure. We say that two area measures of the same dimension m are conformai if they are constant multiples of each other. This equivalence relation partitions the area measures into 26 BRICKELL equivalence classes which are called conformai area measures of dimension m. It is clear tha the definitions of transversality and angular metric are valid for a conformai area measure. We restrict ourselves to symmetric measures so that transversality can be regarded as a function between Grassmann manifolds of non-oriented planes. Also the angular metrics can be regarded as Riemannian metrics on these manifolds. The proof of the following theorem is contained in R ef. [5]. Theorem 2. Let ¡/’i, ¿V i denote the sets of symmetric conformai area measures on V of dimensions 1 and n-1 respectively. Then there is a bijection *-» such that the corresponding transversality functions are inverses of each other and are isom etries of the corresponding angular m e tric s . Presumably, it would not be difficult to extend this theorem to measures of arbitrary dimension. 2. ONE-DIMENSIONAL MEASURES In this paragraph we discuss two theorems which apply in the special case m = 1. For brevity we call a symmetric area measure of dimension 1 a differentiable norm. Note also that Gj n, the manifold of directions in V, is diffeomorphic to the standard (n-l)-spfiere. Let us suppose that n>2. A calculation shows that if the norm L is defined by a scalar product on V then its angular m etric has constant curvature equal to 1. Converse^ we have^-S], Theorem 3. Suppose that L is a differentiable norm on a real vector space V of dimension >2, and that its angular metric has constant curvature equal to 1. Then L is derived from a scalar product on V. We make some remarks about the proof of this theorem. Let us choose a scalar product on V and denote the corresponding norm by L. It follows from (Ref. [9], Chapter 6, Theorem 7^_10) that there is an isom etry Gj n->- Gj n between the angular m etrics of L and L. Theorem 3 can be proved by showing that the isometry is the projection of a linear transformation V - 0 -» V - 0, and this is essentially what is done in Ref. [6]. The method raises the following question which makes sense for measures of any dimension. First note that any non-singular linear transformation V-» V induces, in an obvious way, a diffeomorphism Gmin-> Gm_n. Now suppose that f: Gm n-> Gm_n is an isometry between the angular m etrics of two area measures of the same dimension. Is f necessarily induced by a linear transformation V-*V? The second theorem we wish to mention is due to Schneider [7]. Theorem 4. Suppose that L is a differentiable norm on a real vector space of dimension n and let WL be the volume of Gj n in the angular m etric of L. Then where С,,.! is the volume of the unit (n-1 )-sphere, and the equality holds if and only if L is derived from a scalar product on V. IAEA-SMR-11/10 27 Theorem 3 can be deduced easily from Theorem 4. For if the angular metric of L has constant curvature 1 then G1>n with this metric is isometric with the unit (n-l)-sphere (Ref. [9], Chapter 6, Theorem 7. 10) and, con sequently, WL = Cn_i. We do not know if the angular m etric defined by a scalar product has a corresponding extremal property in dimension m >l. Another question is whether the symmetry condition imposed on L in Theorem 3 is necessary. It is known that it can be omitted provided that V has dimension 3 and L is analytic [8]. Theorems analogous to Theorems 3 and 4 are true for measures of codimension 1. They can, e. g. be deduced by means of Theorem 2. 3. SCALAR PRODUCTS ASSOCIATED WITH REGULAR AREA MEASURES Let S(V) denote the manifold of all scalar products on V. A differentiable function Gm n -*■ S(V) will be called an S-map. Given a regular area measure L of dimension 1 we can define an S-map Gi n S(V) in the following way. Choose a basis [E1; . . . , En] of V, write p1 for p* and put , , 92(iL2), . . , ^ (*> = э 1*3 = 1...... n where e is any non-zero vector in тг. The matrix [gij(тг)] can be shown to be positive definite and therefore a unique scalar product <( , )> is deter mined by the conditions < E j, E j > = g ij(7r) The scalar product is independent of the chosen basis. We pause to mention the following theorem due to Deicke [10,11]. Theorem 5. Suppose that the determinant of the matrix [g¡j ] is constant on GitD. Then the matrix [gy] is constant and therefore the measure L is derived from a scalar product on V. An S-map Gn-i>n S(V) can be defined in the case of codimension 1, again by a simple formula [12]. Theorem 5 is still valid in this case. Because of the fundamental importance of these ideas in the work of Cartan [12,13] it is natural to try and define an S-map for an area measure of arbitrary dimension. It is essential that the definition be intrinsic, that is, an automorphism of V which transforms a measure L into a measure L must also transform the corresponding S-maps into each other. We will describe one way of dealing with this problem. Let G denote the group SLmxLn_m, so that an element of G is an ordered pair (r,s) where r is an mxm matrix with determinant 1 and s is a non-singular (n-m)x(n -m ) matrix. Let 38 denote the manifold of positive definite nXn matrices of the form h 0 0 к 28 BRICKELL where h is of order mxm and has determinant 1. G acts on 33 on the right by 'h 0 1 rThr 0 _0 к J 0 sTks _ Put N =m(n-m) and let denote the manifold of positive definite NxN matrices. We shall write X = [X“b6] for a typical element, the rows and columns of X being indexed by the pairs , or, ,8 = 1 . . . . , m; a, b = m+1, . . . , n. G acts on & on the right by X ^ X^r?r,®£ X, 11, C, d ,-1 Suppose now that L is a regular area measure of dimension m. We fix a plane TrGGm n and consider those n-fram es b = [bj, . . . , bn] of V such that b1; bm are in 7Г, L(bi, bm)=1 and b m + 1 1 bn are in the (n-m)-plane transversal to w. The group G acts on the right on the union of these frames and, as n varies, we get a principal bundle В over Gm_n w ith group G. Suppose that x: В SB is an equivariant map, that is x(bg) = x(b)g for geG. Then, given 7reGm n , we can determine a scalar product on V by the condition [ b„ + I < b a, a = 1 , , m ; a = m + l , . . . , n determine a co-ordinate neighbourhood of 7r with co-ordinate functions u^. The frame X(b) is the m(n-m)-frame f of vectors f“ where IAEA-SMR-11/10 29 We put С = A(B). The m ap X is equivariant with respect to the action of G on С defined by f=[f?]- [ X f№ а,ь The angular m etric, which is a Riemannian metric on Gm n , enables us to define a further equivariant map p: by f - [ x S l w here =<^fa< fb^ Finally let г: á®-» SB be a differentiable equivariant map. The composition r » p » X: B->^ is equivariant and can be used to define an S-map intrinsically associated with the regular area measure L. Of course, we still have to prove that such equivariant maps r: &>-* SB exist. Ref. [14] contains an existence proof for a general value of m but the method certainly does not lead to any explicit algebraic formula. It is not difficult to see why there are simple formulae for S-maps in the special cases m = 1 and m = n - 1. For in these cases G acts transitively on and there are unique equivariant maps which take the unit matrix in to the unit matrix in^. One way of studying area measures is through the orbit structure of p(C). If m = 1 or n - 1 then p(C) is the orbit containing the unit matrix in Sfi and it is natural to ask for the other measures which have this property. The answer, due essentially to Tandai [15], is that they are just the measures derived from a scalar product on V. We will lead up to a proof of this fact by considering local sections of the bundle B. We rem ark that Ref. [16] contains a more detailed study of this bundle. L et b = [bj, . . . , bn] be a local section of B. Choosing a fixed basis [Е1; . . . , En] for V we can express b in term s of this basis as i, j = 1, . . . , n Í We write Q for the matrix [qj ] where i and j index the rows and columns, respectively, and introduce the matrix of differential forms П =Q'1dQ. The matrix fi is independent of the chosen basis for V and satisfies the re la tio n dQ + Q A fi = 0 (3. 1) where л denotes the exterior product. The following calculations relate П to the area measure. Let [jujlj] be the dual basis to X(b). A short calculation shows that ^ q? dqjj, <2 = 1 , ...,m; a=m+l,...,n j where qj1 are elements of the inverse matrix Q"1 = [qj]. The angular metric can, of course, be expressed in terms of the differential forms jj“. 30 BR1CKELL To do this we need the identity pj, = 0 (3.2) J which can be derived from the identity (1. 2). We write dqj*= X 61 dq“= Z ( Z +Z 4e^ ) dq“ = ^^a+^e^fdqa j j а 8 a j. В It follows by using the identity (3. 2) that the angular m etric £ Híf(e)dq¿dq]s= £ Х“®мУ6 a.6,i,j ot,B, a,b w here Kb = XH^e)qa4b i.i and e denotes the m-frame [bi, ..., bm]. The identity (1.2) and the transversality condition on the frames in В together imply that — a 1 T a, * q t = rLi(e) Consequently, a calculation shows that ^ qfdqà = - ^q'adqf = - ^ Xfbn\ i i 8,b Finally, it follows from the condition L(e) =1 that ^ q “d q « = о The preceding calculations have the following consequences. The m a trix в v S3 = Ф w here ц is the matrix [ /4J and в, ф, v are matrices of differential forms with trace 0 = 0. The angular metric is - trace (/1• v) w h ere ц • v is the symmetric product. We use these ideas and notations to prove Theorem 6. If the area measure is derived from a scalar product on V then p(C) is the orbit containing the unit matrix in The converse state ment is true provided that 2 SmSn - 2. IAEA-SMR-11/10 31 Proof. Suppose that the area measure is defined by a scalar product on V. We choose the local section b of В and the fixed basis [Ej, . . . , En] to be orthonormal. Consequently, the matrix Q is orthogonal, Q is skew- symmetric and therefore the angular metric is 2 ^ follows that ot, a the basis f = A.(b) is orthonormal and that p(f) is the unit matrix in There fore, p(C) is the orbit of £0 which contains the unit matrix. Conversely if p(C) is this orbit then there is a unique equivariant map t : p(C)-+3§ which takes the unit matrix of ¿0to the unit matrix of SS. We use the composition т ° p ° À to define an S-map Л: Gmtn ^ S(V), and we shall complete the proof by showing that if 2 SmSn- 2 then A is constant. To do this, we choose the local section b so that Ь(гг) is orthonormal with respect to the scalar product A(ir). It follows that the angular metric is 2(^â)2 and therefore а, а L et [Ej, .. . , E n] be a fixed basis for V and denote the matrix of scalar p ro d u cts KE¡, Ej )>] by K. A calculation shows that dK = - (Q '1 )T ( Г2 + f2T) Q '1 and, consequently, we need to prove that Q is skew-symmetric. It follows from relation (3. 1) that d/u + /uA6+0A/u = O, d;uT + 9 Л д T + д т A 0 = 0 We transpose the second equality and subtract it from the first to obtain M A (0 + 0 T) + (ф + ) А ц = 0 We omit the elementary algebra which shows that if 2 im án - 2 then this relation, together with the fact that в+вт has zero trace, implies that 0 +0^ =0, ф + ф^ = 0. 4. AREA MEASURES ON MANIFOLDS A regular area measure on a differentiable manifold M of dimension n is a differentiable assignment of a regular area measure to each tangent space Tp(M), p£M. We suppose that the measures have dimension m and denote the manifold of oriented tangent m-planes to M by ¡Й. Let a: M-»M be the natural projection. The work in section 3 implies that there exists a scalar product in the induced vector bundle a ‘1(TM) which is intrinsically associated with the area measure. An important consequence is that we can attempt to construct a connection in cr^TM) by a procedure analogous to that used to construct the Riemannian connection in Riemannian geometry. The procedure is successful in the case m = 1 [13] and leads to a solution of the local equivalence problem. 32 BRIC KELL For m > l the pseudo-group of local automorphisms can be infinite dimensional and it is necessary to impose restrictions on the area measure. We refer to Ref. [12] for details of the case m = n - 1. In Refs [14, 3] the scalar product is used in a different way to provide answers to some simple global questions about area measures on a manifold. The formula for the scalar product in the case m = 1 has been generalized in a different direction by Iwamoto [17]. For a regular area measure of arbitrary dimension m he has given an explicit formula which associates with each 7reGm n a scalar product on the m-vectors of V. We refer to Refs [18,19] for an approach to the theory of area measures which makes use of Iwamoto's work. REFERENCES [1] DELENS, P., La métrique angulaire des espaces de Finsler, Actualités Sci. et Ind. No. 80 Paris (1934). [2] GOLAB, S., Sur la représentation conforme de deux espaces de Finsler, C.R. Acad. Sci. Paris 196 (1933) 986. [3] BRICKELL, F., AL-BORNEY, M.S., A note on area measures, J. London Math. Soc. 4 (1972) 466. [4] M ORREY, C.B . , Multiple Integrals in the Calculus of Variations, Springer-Verlag, N e w York (1966) 61. [5] BRICKELL, F., A relation between Finsler and Cartan structures, Tensor, N.S. 2J3 (1972) 360. [6] BRICKELL, F., A theorem on homogeneous functions, J. London Math. Soc. 42 (1967) 325. [7] SCHNEIDER, R., Über die Finslerrâume mit Sjj^^ 0t Arch. Math. 19 (1968) 656. [ 8] M Ü N Z N E R , H. F., Die Poincarésche Indexmethode und ihre Anwendungen in der affinen Flachentheorie, Dissertation FU Berlin (1963). [9] K OBAYASHI, S., N O M I Z U , K . , Foundations of Differential Geometry, _1, N e w York and London (1963). [10] DEICKE, A., liber die Finslerrâume mit A¿=0, Arch. Math. 4 (1953 ) 45. [11] BRICKELL, F., A new proof of Deicke's theorem on homogeneous functions, Proc. Amer. Math. Soc. 1£ (1965) 190. [12] C A R T A N , E., Les espaces métriques fondés sur la notion d’aire, Actualités Sci. Ind. No. 72, Paris (1933). [13] C A R T A N , E., Les espaces de Finsler, Actualités Sci. Ind. No. 79, Paris (1934). [14] BRICKELL, F., Differentiable manifolds with an area measure, Can. J. Math. Г9 (1967) 540. [ 15] TANDAI, K . , On areal spaces VI. Tensor (N. S. ) 3 (1953) 40. [16] W A G N E R , V . , Geometry of a space with an areal metric and its applications to the calculus of variations, Rec. Math. 19 (1946) 341 (in Russian). [17] I W A M O T O , H . , O n geometries associated with multiple integrals, Mathematica Japónica ¿(1948) 74. [18] DAVIES, E. T. , Areal spaces, Ann. Mat. Pura Appl. 4 55 (1961) 63. [19] K A W A G U C H I , A . , O n the theory of areal spaces, Bull. Calcutta Math. Soc. 56 (1964) 91. IAEA-SMR-11/11 THE DIFFERENTIABILITY OF TRANSFORMATIONS WHICH PRESERVE GEODESICS F. BRICKELL Institute of Mathematics, University of Southampton, United Kingdom Abstract THE DIFFERENTIABILITY OF TRANSFORMATIONS WHICH PRESERVE GEODESICS. The aim of this paper is to study under which extra conditions a bijection f of a smooth Riemannian manifold M which preserves geodesics is a diffeomorphism. The theorem given here (with proof) is: If f is a homeomorphism and dim M > 2 then f is a diffeomorphism. The arguments involve only basic ideas in differential geometry. This contribution is based on a paper published some time ago in Proc. Am. Math. Soc. J_6 (1965) 567-574. The topic is reasonably general and the arguments involve only basic ideas in differential geometry. In addition, it seems likely that the result could be improved. We shall begin by describing two theorems in each of which a simple geometrical condition on a transformation is shown to have a surprisingly strong consequence. Let Vn denote a real vector space of dimension n, and say that non zero vectors u, veV n are equivalent if u = Xv for some non-zero real n u m b e r X. An equivalence class is called a non-oriented direction in Vn but in this contribution we shall omit the word non-oriented. The set Pn_1 of all directions in Vn is called real projective space of dimension n-1. A subset of directions corresponding to the non-zero vectors in a two-dimensional subspace of Vn is called a line in p n_1 . ^ A non-singular linear transfórmation ф: Vn -*■ Vn induces a bijection ф: p n_1 -» p n_1 which takes lines into lines. Suppose, conversely, that f: Pn 1 -* Pn 1 is a bijection which takes lines into lines. Is f induced by a non-singular linear transformation of Vn ? Of course, this is not so if n = 2 for then Pn_1 = P 1 consists of just one line and any bijection pre serves this line. However, for n > 2 it can be proved that f is induced by such a transformation, and this fact is known as the fundamental theorem of real projective geometry. The second theorem which we wish to mention is a theorem in Riemannian geometry. Let M be a C“ Riemannian manifold. Then M can be made into a metric space and we denote the distance function by p. A bijection f: M -» M is said to be an isom etry if p(f(x), f(y)) = p(x, y) for all x, yeM . The theorem of Myers-Steenrod states that an isometry of a Riemannian manifold is necessarily a diffeomorphism. In what follows we shall also be working with a C” Riemannian manifold M and it will be important to bear in mind the distinction between a geodesic curve and its range. We shall use the word geodesic to mean the range 33 34 BRICKELL of a geodesic curve. Furtherm ore, a bijection f: M -* M will be said to preserve the geodesics of M if, for any maximal geodesic gCM, both f(g) and f -1(g) a re a lso m a x im al g eo d e sic s of M. Having established this notation we can state the main aim of this paper. It is to consider the following question which is related to both the previous theorems. Let f be a bijection of the C“ Riemannian manifold M which preserves the geodesics of M. Is f necessarily a diffeomorphism? To start with we make some remarks which are intended to throw light on the problem. A real projective space admits a natural C" structure and a Riemannian metric for which the lines are the maximal geodesics. For this particular case, the fundamental theorem of real projective geometry implies that the answer is yes, provided that the dimension of the space is > 1. On the other hand, as was pointed out to me by T. Poston, it is very easy to construct bijections of the unit sphere SncR n+1 which preserve the great circles and which are not homeomorphisms. For example, let us define x = (xi, ..., xn+1)e Rn+1 to be rational if all the numbers x¿ are rational and to be irrational otherwise. Then f: Sn -» Sn given by f(x) = x, for x rational, f(x) = -x for x irrational is such a bijection. The last example shows that some restriction on f is necessary if it is to be proved to be a diffeomorphism. And, of course, the dimension of M must be > 1. With these remarks as a background we state a partial answer to the question in the following theorem. Theorem. Let M be a C" Riemannian manifold of dimension > 2 and let f: M -* M be a homeomorphism which preserves the geodesics of M. Then f is a diffeomorphism. The restriction dim M>2 which appears in this theorem in place of dim M > 1 is irritating as we do not think that it is necessary. However, it is pleasing that the theorem of Myers-Steenrod is a simple corollary. For an isometry is a homeomorphism and it is not difficult to show that it preserves geodesics. In the proof of the theorem we shall make use of special neighbour hoods for the points of M which we call С-neighbourhoods. To describe these neighbourhoods we consider a point xeM and the exponential mapping exp* of the tangent space Tx into M. A normal neighbourhood of x is e x p xW w h ere (i) exp is a diffeomorphism on the open set WcTx, (ii) if weW then tweW fo r a ll t, 0 s t S 1. A С-neighbourhood U of x is a neighbourhood which is a normal neighbour hood of each of its points. Thus any two points of U can be joined by a unique geodesic segment lying in U. It can be shown that every neigh bourhood of x contains a С-neighbourhood of x. The proof proceeds in five steps which we shall outline. First note that a geodesic through a point x determines a unique direction in the projective space Px associated with Tx . For although the geodesic is the range of an infinite number of geodesic curves, the tangent vectors to IAEA-SMR-11/11 35 these curves at x are all non-zero scalar multiples of each other. Con sequently, as the homeomorphism f preserves geodesics, it determines in an obvious way a function fx: Px Pf(x) fo r e a ch xeM. It is not difficult to show that fx is a homeomorphism. The second step is to establish the following technical lemma. It is an exercise in the use of the exponential map. Lemma. Let U be a С-neighbourhood of xeM and let a, b be geodesics through x in U. Suppose that an, bn are sequences on these geodesics such that lim an = lim bn = x. Let cn be a point on the unique geodesic in U joining an to bn, and denote by yn the direction at x of the geodesic in U joining x to cn. Then if lim yn exists this limit lies on the line in Px n-> » containing the directions of a and b at x. Further, every direction on this line can be obtained as such a limit. The next step is to use the lemma to prove that fx takes lines into lines. To do this choose С-neighbourhoods U, V of x, f(x) respectively such that f(U)CV. Then, given any two directions a, /3 in Px , let a,b be the geodesics in U with these directions. Let у denote a direction on the line in Px containing a, (3 and, in the notation of the lemma, construct sequences an, bn, cn such that lim yn = y. The direction fx(7n) is the n-» " direction at f(x) of the geodesic joining f(x) to f(cn) in V, and, from the continuity of fx, lim fx(Yn) exists and is equal to fx(y)- Consequently, n-* " the lemma may be applied to the sequences f(an), f(bn), f(cn) in V to show that fx(y) lies on the line in Pf(X) containing fx(o') and fx((3). It follows from the fundamental theorem of real projective geometry that, if dim M >2, then fx is induced by a non-singular linear transformation between Tx and Tf(Xj and is therefore a diffeomorphism. The fourth step is concerned with a local function which is like bipolar co-ordinates. Let U be a С-neighbourhood and fix two points y, z in U. Given a point xeU (f y, z), there are unique geodesics in U joining x to у and x to z. These geodesics determine directions in Py , PZ! respectively, and consequently a function Tyz : U - { y, z} - Py XPZ can be defined in an obvious way. It is not difficult to show that T yz is an immersion at all points which are not on the geodesic in U containing у and z. The last step is to show that f is differentiable at a general point xeM. To do this choose С-neighbourhoods U, V of x, f(x) respectively su ch th at f(U)cV. Fix points y, z in U such that x is not on the geodesic in U containing у and z. Then on some neighbourhood of x, rYZ ° f = (fyX fz)°Ty2 36 BR1CKELL where Y = f(y), Z = f(z). It follows that the composition tYz, " f is differentiable on this neighbourhood and therefore, since f is continuous and r yz ks an immersion, f also is differentiable. A sim ilar argument applies to f"1 and so f is a diffeomorphism. It is perhaps worthwhile pointing out that the arguments we have used work in more general situations. For example, they apply to the geodesics of any spray on a differentiable manifold. IAEA-SMR-11/12 STABILITY THEOREMS FOR R2-ACTIONS ON MANIFOLDS C. CAMACHO Instituto de M atem ática Pura e A plicada, Rio de Janeiro, Brazil Abstract STABILITY THEOREMS FOR R2-ACTIONS ON MANIFOLDS. This paper introduces the notion of hyperbolicity for fixed points of an R2-action on a manifold and gives interpretation of this definition in terms of transversality. Hyperbolicity is a necessary condition of the stability of the germ of the action and it is also sufficient for lower dimensional manifolds. A family of R2-actions generalizing the Morse-Smale flows is also defined. These actions exist on 2-manifolds and Grassmann manifolds; moreover they are ^-stable. An action of a Lie group G on a manifold M is a homomorphism tp: G ->• Diff(M) of G into the group of diffeomorphisms of M. This action X(x) = A ^(x)/t = о (*) T hus tpt(x) can be considered as the solution of the ordinary differential equation (*) with initial condition tp0(x) = x and the orbit through x coincides with the integral curve of X passing through x. Conversely given a differentiable (say C1) vector field X on a compact manifold M, the existence and uniqueness theorem for differential equations and the compactness of M guarantee the existence of a flow on M satisfying (*). Here we shall be concerned with actions tp of the group G = R2, i.e. th e direct sum of two copies of R. By restricting tp to each of these copies one obtains two flows, say f and rj satisfying Çs rjt = r?tÇs for all s, t. Let X and Y be the vector fields generated by f and r). Then the commutativity condition on the flows means that the Lie bracket of X and Y is identically zero; one then says that X and Y commute. By looking at the closed subgroups of R2 one obtains all possible orbits of tp. If dim 0 x(tp) = 0 th en 0 x(tp) is a point. If dim & x(cp) = 1 , 0 (tp) is homeomorphic to S1 or R. If dim & x{tp) = 2, 0 x(tp) is homeomorphic to S1 X S1, S1 X R o r R 2. 37 38 CAMACHO Example 1. The Reaction on the plane generated by two commuting vector fields X(x i< x 2) = (Xj+x2, Xg) and YfXj.Xg) = (Xj, xg) has the following orbit structure: the half planes x2 > 0 and Xg < 0 are 2-orbits, the half lines = 0, Xj < 0 and x2= 0, xx > 0 are 1-orbits, the origin is a fixed point. Example 2. The Reaction on 3-space given by XJXj, x2, x3) = (-x2,x i.x3) and Yfx^x^x) = (x^x^O) has the following structure. The origin is a fixed point. The half lines xj = X2 = 0, хз> 0 and x: = x 2= 0, x3 < 0 are 1-orbits. The plane XjX2 minus the origin is a cylindrical orbit. All other orbits are homeomorphic to R2 with set boundary {x3 = 0 } U { Xj = x2 = 0 }. The most simple kind of actions are the linear ones, that is, homo- morphisms p : G ->■ Aut(E) of G into the group of linear automorphisms of a vector space. Both examples above are linear. Hyperbolic linear actions. Let G be isomorphic to Z or R. Then p is called hyperbolic if for s f 0 all eigenvalues of p(s) are different from one in absolute value. Suppose now G = R2then p is called hyperbolic if: (i) There exists a p-invariant splitting of E, E = œ Et such that for any t, p is transitive on each connected component of Et - 0. (Notice that (i) readily implies that dim Et = 1 or 2. ) Let ir : E -> œ E t' tVt be the projection map. For any non-zero v 6 Et the map xt : Gy(p) -» Aut ( ® E .) defined a s Xf(g) ° 7r = 7r ° p(g) is by (i) the action of a group tV t 1 isomorphic to Z or R. (ii) For each t xt is hyperbolic. Example 3. Example 1 is not hyperbolic; however, a nearby action generated by X£( x i,x 2) = (Xj+Xg+eXj, x 2) and Y(x1,x 2) = (x^ xg) is hyperbolic. More generally: Theorem: Hyperbolic linear Reactions form an open and dense subset of the space of linear Reactions. There is also an interpretation for hyperbolicity in term s of trans versality given in the following Theorem: Let E с Aut(E) be the set of non-hyperbolic automorphisms of a vector space E (i.e. A E E if some eigenvalue of A lies on the unit circle). Then p : R2 -» Aut(E) is hyperbolic if and only if p intersects E transversally out of the identity. Assume from now on that M is compact. Let xr(M) be the space of Cr vector fields with a Cr norm || .. . . || . A topology is introduced in the space of actions as follows. Call T^ : R2-» xr(M) the Lie algebra homomorphism induced by An ac tio n (p is called structurally stable if there is a neighbourhood N( FIG. 1. Linear examples on Euclidean 3-space. 40 CAMACHO Theorem: Let x be a hyperbolic fixed point of an action Diff(M) such that dim M £ 4. Then the germ of gis a diffeomorphism contracting at x. Then the germ of FIG. 2. Orbit structure given by theorem in text. IAEA-SMR-11/12 41 Indeed given a linear action p on En and the canonical transitive action ■iof Aut(En) on the Grassmann manifold Gm, n-m one obtains Reactions p on Gm,n-m by the following diagram: R A ut(E n) \ ■ф \ \ Diff(Gm, n-m) One proves that under certain conditions on the eigenvalues of p, p satisfies (i)-(ii)-(iii)- As an illustration we describe an R -action on S3 given by this theorem: L et p : R2 -» Aut(R4) be defined by písj, s2) = ex p ^ X ^ s 2X 2) w here Xj = diag(X, X, X 3, X4) X4 < X < X3 and X2 = diag(,u, M2. M3, M ) M3 < U < M2. The orbit structure is as follows. The intersections of the x¡-axis with S3 yield eight hyperbolic fixed points of "p. The intersections of the x¡Xj- planes of R4 w ith S3 give the set of non-wandering points of ~p. So Q(p) = 6 U Г2 where each S3, is a 1-sphere, t = 1 t t ^ Suppose Í21; Í32, £33are the 1-spheres passing through the fixed points of "p lying on the xj-axis. The intersections of the 3-planes XjXjX^ of R 4 with S3yield four p-invariant 2-spheres. Out of these spheres all orbits of ~p are homeomorphic to R 2 with boundary Í34 U Q5 U Щ. This example as well as the examples on 2-manifolds are structurally stable. In fact one has the following theorems: Theorem. Reactions on a 2-manifold satisfying (i)-(ii) are structurally sta b le [ 3 ]. о Theorem. R -actions on 3-manifolds induced by two commuting gradient vector fields and satisfying (i)-(ii) are structurally stable. The following £3-stability theorem for Reactions is a generalization of a theorem of J. Palis for diffeomorphisms (see Ref. [4] ). Theorem ([ 3] ). Reactions on an n-manifold satisfying (i)-(ii)-(iii) are Í3-sta b le . REFERENCES [1] C A M A C H O , C . , On Rk X Z'^-actions, Proc. Symp. Dynamical Systems, Salvador, Brazil (1971). [2] SMALE, S., Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967) 747-817. [3] C A M A C H O , C . , Morse-Smale Reactions on 2-manifolds, Proc. Symp. Dynamical System, Salvador, Brazil (1971). [4] PALIS, J., On Morse-Smale dynamical systems, Topology, _8 (1969) 385-405. [5] C A M A C H O , C . , A n instability theorem for Reactions to be published. IAEA-SMR-11/13 INTRODUCTION TO MINIMAL-SURFACE THEORY E. DE GIORGI Scuola Normale Superiore, Pisa, Italy Abstract INTRODUCTION TO MINIMAL-SURFACE THEORY. Definitions of Hausdorff measure and of perim eter are given, followed by applications to a generalized problem of minimal surfaces. The aim of this contribution is to provide in an informal way some concepts about minimal-surface theory. We first consider the following "Dirichlet problem": Let Bj be the unit sphere of IRn, i. e. Bj={xE ]Rn: | x| < 1} . Let f be a real-valued function defined on the boundary of В^ЭВ^; we suppose that f has a certain degree of regularity (for example, C^BBj)). The problem is thus the following: find a function g G C1(B1) such that g G C°(B1)J g = f on SBj and i / ( l + |V g|V dx g ^ ( l + |Vh|2) dx (1) Bi Bi for any h G C1(B1) П C°(B1) such that h = f on 3Bj. It has been proved that this problem has a unique solution, and that this solution is analytic in Bj. One could also weaken the conditions on f, just requiring that f be continuous. Furthermore, the theorem does not only hold true for a sphere, but also for any open domain (with smooth boundary) with the boundary having positive mean curvature. For these results, see a very clear exposition in Ref. [3]. We must now rem ark that if we remove the condition that the boundary of the domain has positive mean curvature, we can no longer expect to find a solution which is continuous up to the boundary. The following simple problem shows what happens: Let D = {x G ]Rn: 1 < | x |< 2} . Let f be 0 for | x| =2, and M for | x| = 1. Then, for large M, there exists no solution of the Dirichlet problem for minimal surfaces in the sense described above. It is therefore necessary to generalize the concept of surface, and, of course, the concept of area in order to be able to solve more general p ro b le m s. First of all, we need a suitable (n-1 (-dimensional measure. This will be the Hausdorff measure: for any integer k, OSkSn, fo r any s e t E £L]Rn, we define CO Hk(E) = sup 12 uk inf \ (diam Eh)k; UE.DE, diamE. 43 44 DE GIORGI It is easy to prove that Hk is a Borel regular measure, i. e. all Borel sets of IRn are measurable1 and for any set E there exists a Borel set В Э E such that Hk(B) = Нк(Е). Furthermore, on regular manifolds (C1-manifolds, for example) Hk is the same as the common k-dimensional m e a s u re . Now let A be an open set of IRn; for any function g : As^-R, we define the set E(g) by the following relation: E(g) = |(x, z) G IR X R : x G A, zSg (x)j- (3) We come back to the case in which f = 0 for |x| = 2, f = M for | x| = 1. We choose the open set A such that AD D = {x G IR11: 1 S | x| S 2} , and we try to m in im iz e Hn(8E(g) П A) when g = 0 for x G А П -|x G ]Rn: | x | ë 2 g = M for x G A Pi -j x G IRn: lx l Si It is clearly a reasonable generalization, but the Hausdorff measure is not too suitable if we want to use the direct method of the calculus of variations. We shall therefore give a different definition of "perim eter" of a (Borel) set of IRn. Let A be an open set of IR11. For any Borel set В £L IR11 we define the perim eter of В with respect to A by the following relation P (B , A) = sup J div g dx; g G [cJ(A)]n, | g | Ê lj- AnB If P(B, A) < + oo we say that В has finite perim eter with respect to A. If P(B,K) < + со for any open set К such that К is a compact set contained in A(KCCA) we say that В has locally finite perimeter with respect to A. Now for any Borel set В let B1/2 denote the set2 x G IR0 : lim [u p11]’1 meas [A (x) П B] = 1/2 p—* 0 nP if the set В has locally finite perim eter with respect to A, then Р(ВЛА) = Нп_х(В1/2 П A) 1 A set В is said to be measurable with respect to an exterior measure о if for any subset E of the space it holds that о(Е) = о(ЕПВ) + o(E-B). 2 Ap(x) ={y eR n : I y - x I In particular, if В has a C1-boundary, then P (B , A) = (ЭВ П A) The following two theorems allow us to use the direct method of the calculus of variations in problems where the perim eter is involved: 1. Let A be an open set of IRn. {Bj,} is a sequence of Borel sets. If for any set KCC A there exists a constant 7 (K) such that PÍB j^K ) á y (K) fo r any h then there exist a subsequence Bh and a Borel set В such that к ' Bh^к B 111 L loc(A ) (i.e. for any К CCA lim m e as Bu - В U В - B, П К 0 ) k-»°° uk /J 2. If a sequence of Borel sets {B^} tends to a Borel set В in L* (A), then P(B, A) § lim inf P(B , A) h->» There are many different ways of generalizing the minimal-surface problem. The use of perimeter is treated in Ref. [2]; for other methods compare Ref. [1 ] . Further references can be found in Refs [1-3]. REFERENCES [1] ALMGREN. F.J. Jr.. talk at present Summer College. [2] DE GIORGI. E., COLOMBINI, F ., PICCININI, L .C ., Frontière orientate di misura m inim a e problem i connessi, Edizioni Scuola Normale Superiore (Pisa 1972). [3] MIRANDA. М .. Paper to appear in the "Lecture Notes of the CIME Session on Geometric.Measure Theory and Minimal Surfaces". IAEA-SMR-11/14 ON COMPLEX VARIETIES OF NILPOTENT LIE ALGEBRAS (AFTER G. FAVRE) P .DE LA HARPE Mathematics Institute, University of Warwick, Coventry, Warks, United Kingdom Abstract ON COMPLEX VARIETIES OF NILPOTENT LIE ALGEBRAS (AFTER G. FAVRE). It is well-known that there are uncountably many isomorphism classes of nilpotent Lie algebras; hence one expects classification results (if any) to be given in terms of continuous parameters. This paper gives examples proving that such (partial) results can indeed be obtained in a few cases. Sections 1 to 4 illustrate, through nilpotent Lie algebras of maximal rank, the methods of Favre. This is a preparation for the two examples of section 5, where families of nilpotent Lie algebras are parametrized by varieties; these are quotients of projective spaces by finite (possibly trivial) groups of transformation. 1. MOTIVATIONS All Lie algebras considered here are defined over the complex field C, and they are of finite dimension with the only exception of the free algebra Lie(w) and of its ideals appearing in example 6. The set of strictly positive integers is denoted by N* = {1,2,...}. Let _g be an arbitrary Lie algebra. It is a classical result that £ can be written as a sem i-direct product jj = rx,,¿ where ф is an homomorphism from _s to the Lie algebra Der(r) of derivations of jr, where the derived ideal of £ is nilpotent and where s is a direct product of simple Lie algebras: к _s = П s¡ (see Bourbaki [1(1, 5 and 6)] and [2]). i = l Much is known about semi-simple Lie algebras such as s¡; in particular, their full classification has been achieved by E. Cartan (a fast introduction to this theory is provided by Samelson [3]). We shall retain the two following fac ts: (1) There are countably many isomorphism classes only of semi-simple Lie algebras. k (2) Any semi-simple Lie algebra _s is a direct product _s= П s¡ as above and is completely characterized (up to isomorphism) by a finite set of discrete parameters (for example: the number к in the product above, and the rank and type of each of the Si ). In comparison, very little is known about the structure of nilpotent Lie algebras; in particular, a full classification seems to be out of reach at present. However, some results have been recently obtained by various authors including Amiguet [4] and Favre [5]. A partial report on their 47 48 DE LA HARPE work limited to the study of nilpotent Lie algebras _g of maximal rank, as defined in section 3, is given in this paper. (References [4, 5] obtain, however, more extensive information both on £ and on Der(g), and without the restriction of maximal rank. ) 2. SYSTEMS OF WEIGHTS L et £ be an arbitrary Lie algebra and let Der(g) be the Lie algebra of its derivations. A maximal torus for £ i s by definition a subalgebra Tjg of Der(g) which is abelian, which consists of semi-simple derivations (i. e. the linear map Д :£-* £ can be diagonalized for each ДеТ^), and which is maximal among subalgebras of Der(g) having these two properties. Let Tj[ be a fixed maximal torus for £ and let T*jj be its dual (as vector space). Consider the natural representation of Tg in _g, which is induced by the evaluation map Der(g) Xj*-> _g. For each eigenvalue aeT*g, let S°' be the eigenspace jxe_g/A(X) = a(A)X for all AeTgj Let R_g(Tj[) be the (finite) subset of those <* for which dim(gf )> 0. As the elements of Tg form a commuting family of diagonalizable endomorphisms of the vector space £, the algebra £ can be written as £= 0 £ a aeR g(T g) By definition, the system of roots defined by Tg is the set Rj?(Tj?)={a'eT*g/dim(gtI)> 0}; the system of weights defined by Tg is the set Pg(Tg) = {(a, da)eR£(T£)X N*/d« = dim(g“)}. It happens that these systems do not depend essentially on the choice of Tg. More precisely, let £ and ¿ be two Lie algebras with maximal tori Tj[ and T 'g1 . and let Rj[(Tj[) and Rg1 (T'g1) (resp. Pj*(Tj;) and Pg'(T'g')ibe the c o rresp o n d in g sy ste m s of ro o ts (re sp . w eights); then P_g(Tg) and Pjf'(T'¿) are said to be equivalent if there exist linear bisections cp : Tg -> T'g' and ф :£ such that the diagram TgX£ ------£ Ф X ф j ф T T '¿ _ X £ l ------commutes (otherwise said: if the representations of Tg in £ and of T'g' in £l are equivalent). But now, in case g = g' : Proposition 1. Let Tg and T'_gbe two maximal tori for _g. Then there exists an automorphism в of £ such that 0(Tg) = T'jg, where V is the restriction to Tjj of the automorphism Der(_g)------► Der(_g) A '------в-А-в' 1 IAEA-SMR-11/14 49 Proof: It can be read out of a result due to Mostow [6 , (theorem 4.1)], and can also be found in Amiguet's thesis [4 (4, theorem 16)] or Favre's th e s is [7 (1, th e o re m 2)]. It follows from proposition 1 that, keeping the same notations, the d ia g ra m T gX .g ------» £ 0X 0 T > _ g X £ ------► £ co m m u tes, so th at Pjg(Tj[) and P.g(T'_g) a re equivalent. T hey can be _ f Pg(T'jg)------► P£(T£) identified by 0 - , where '■'в: T'*_g-> T*_g is the (o',do') i------*'(t0(al), da1) transposed map of 0. Hence, we will write Pg (resp. Rg) instead of Pg(Tg) (resp. Rg(Tj*)) and call it the system of weights (resp. of roots) of the Lie algebra_g. Example 1. Let _g be a semi-simple Lie algebra. Then Der(g) is canonically isomorphic to £ so th at: (a) maximal tori for _g can be identified with Cartan subalgebras of £; (b) the root system of _g defined above coincides with the root system usually defined. Note that in this case da = 1 for each non-zero root a and da is equal to the ra n k of _g when a = 0. Example 2. Let £ be an abelian Lie algebra. Then Der(g) is the Lie algebra _gl(_g) of all linear maps from _g into itself. Let (e¡)ie Ibe a basis fe¡ if j = к of £ and let (E¡ j)¡ j be the usual maps defined by E; . (ek ) = - o th erw ise. Then T£= ф CEU is a maximal torus for £. For each je I, let ieI ' [Tg = © CEU ------«-C Qfj-e T*g be the map 4 ’ . Then Rg = eT-^g/jel} and EXiEu '------daj = 1 for each j el. Example 3. Let _g be a nilpotent Lie algebra such that every derivation of £ is nilpotent. (Such algebras do exist: see Bourbaki [1 (4, ex. 19 and 5, ex. 13)] or Favre [8]). Then the only maximal torus for £ is {0}, so th a t P_g is a set with one element: P_g= {(0, dim(g))}. Example 4. Let £ be a semi-simple Lie algebra, let jd be a parabolic subalgebra of £ and let _b be a Borel subalgebra of £ contained in £. Both £ and b have inner derivations only and have zero centres (see Favre [7 (4. 4)] and [9]). Knowing that, one can easily describe the systems of weights of £ and of b in term of the roots of £; details are left to the reader (see also example 5 below). 50 DE LA HARPE 3. N ILPO TEN T LIE ALGEBRAS Let_g be a Lie algebra and let (Cn_g)neN=;t be its lower central series, defined by Cx_g=¿ and Cn+1_g= [_g, Cnj?] when ne№ !. Then _g is nilpotent if Cn_g= {0} for n large enough. If _g is a nilpotent Lie algebra, there are three important invariants attached to _g: The type s of g is the codimension of C 2g in The nilpotent class (or simply class, below) is the largest integer p such that CPj;/ {0}. The rank к of ¿ is the dimension of the maximal tori for _g. Proposition 2. Let |b e a nilpotent Lie algebra, and let s, p, к be as above. Then: (a) any complement w of C 2g in ¿ generates £ as a Lie algebra; (b) any s e t of g e n e ra to rs fo r ¿ contains at least s elements; (c) if w is any complement of C 2¿ in two derivations of _g which coincide on w are identical; (d) the in equ ality OS k s s holds. Proof: Each part of proposition 2 is a reasonably easy exercise, left to the reader; alternatively, see Amiguet [4 (3, theorem 9, and 4, lemma 9)]. The general programme now is to apply the notions introduced in section 1 to nilpotent Lie algebras. In order to make life easier, we restrict ourselves from now on to nilpotent Lie algebras of maximal rank, namely to those for which k = s (notations as in proposition 2d). It is shown in Ref. [5] that part (but not all!!) of the results below carry over to arbitrary nilpotent Lie algebras. Note that nilpotent Lie algebras of maximal rank and Lie algebras behaving as that of example 3 are the two extreme cases: к = s and к = 0. We will now state some properties of algebras of maximal rank, and then "prove" them by verification on the examples of the next section. Honest proofs are given in Ref. [5 (I)] . Definition. Let R be a finite subset of a vector space V which spans V, and let В be a subset of R (think of Rg in T*g, with a Lie algebra). (a) В is said to be a basis of R if it is a basis of V in the usual sense and if moreover any element of R can be written as a linear combination of elements in В with positive integer coefficients. (b) Let В be a basis of R; a В-path in R is a sequence (0i, (32 , . . . , (3q ) of elements of R such that, for each ie{l, 2, . . . , q- 1}, e ith e r -|3i+1e B o r /3i+1 - / 3 i e B . (c) Let В be a basis of R; a B-connected component of R is an arc-connected component of R for the previous definition of B-paths. Proposition 3. Let g be a nilpotent Lie algebra of type s, of class p and of maximal rank ; let Tj* be a maximal torus for _g; then the system of weights P_g enjoys the following properties: IAEA-SMR-11/14 51 (a) Rg contains a unique basis В = {a,, . . . ,a } . s s (b) L et a = ^ п;а; e Rg, and let |e | = ^ rijbe its length; then i= 1 i = 1 la |a I S p. For each ie {1, . . . , s} , the only multiple of which is in Rg is Oj itself. (c) There is a function f of s integer variables, taking integer values, s and such that the following holds: if a = ^ is as in (b), then i = l d a = d im fg” ) 5 ffnj, . . . , ns). (d) Let (Rj_g)j = j { be the B-connected components of Rg and let gj = © . g“ fo r each j e { 1, . . , I }. aeRJ£ Then: each is a nilpotent Lie algebra of maximal rank, say of type Sj and of class Pj ; the equalities i i Sj and p = sup (pj ) i = i j = 1 hold; moreover gj cannot be written as a direct product of two Lie algebras, this for each j e ll, . ., SL}. N. B. : the best possible function f for (c) is given by the dimensions of the root spaces of the ad hoc model (example 6), and can be expressed in term s of the Mobius function. (It is obviously the same function as that in Bourbaki [10 (II. 3, formula 16)). 4. EXAMPLES OF WEIGHT SYSTEMS Example 5. Let g be a semi-simple Lie algebra, let h_be a Cartan subalgebra of g, let á? be the set of non-zero roots of g with respect to h, le t & = 3t*U ¿ft " be a partition ofá?in positive and negative roots, and let r rT = 0 g ? g = n l0 h ® n w ith -i ae.31 I n = 0 £ ? C(Ë^ + be a Cartan decomposition of g. Then n is clearly a nilpotent Lie algebra whose type is the rank of _g. Any element X of h defines a derivation j y i ___►[x Y]’ So that there is an injection h -» Der(n); and it is clear that the image of h by this injection is a maximal torus for n. Hence n is of maximal rank. The proof of proposition 3 for this example follows straightforwardly from the theory of semi-simple Lie algebras; in particular, Rn=¿¿?+ and do = dim(ga) = 1 fo r each a e Rn. (Fig. 1). 52 DE LA HARPE a 1+a 2 FIG.V 1. Rn for g= s 1(3,0). A 3-dimensional hardware model for Rn when _g= so(7, C) looks nice, though useless for soap-bubbles. Example 6. Let us try and construct the nilpotent Lie algebra of type s and class p, which has "as few extra relations as possible". Let w be a vector space of dimension s, let ®w be its tensor algebra, and let Lie(w) be the Lie algebra generated by w in ®w. Though Lie(w) is infinite dimensional as soon as s ë 2, the quotient m(s, p) = L¿e(w)/CI>+1(Lie(w)) is finite dimensional for each p e№ ; it is by definition the model of type s and of class p. (Exercise: show that in(s,p) has indeed type s and class p.) The integers s and p being fixed once and for all, let us write in instead of m(s, p). We identify w with a complement of C2 in in m and we choose a basis (e¡)i=i _ # s of w. For each ie{l,...,s}, let Д ¿ be the derivation of in which extends the linear map w ------*■ in e . ,______► \ ei if J = i J [O o th e rw ise S Then Tm= ф С Д, is a maximal torus for m, and in particular m. is of i— 1 maximal rank. Let (аь . . . ,as) be the basis of T*m which is dual to the basis (Aj, . . ,As) of Tm. Then n¡eN for each i = 1, .., s s S a = ^ njQ-j e T * m 1 § ^ n¡ = O' 1 È p i “ 1 i = 1 the only multiple of a¡ which is in Rm is itself, for each i = 1, . . , s. as one can show by writing a basis of rn in term of the e¡'s and of their b ra c k e ts. Digression. The construction of example 6 is of a style which is fairly standard in many domains. In the particular case of nilpotent Lie algebras of given type and class, it seems to go back at least to Scorza [11]. The derivations of the model have been studied by Schenkman [12]. The IAEA -SM R -11/14 53 model has been used by Sato [13], and by Dyers for constructing his example given in [14]. According to the motivations which introduce example 6, any nilpotent Lie algebra of type s and class p is isomorphic to a quotient m (s, p)/a where a is an ideal in the model such that CP rn(s, p) ^aCC2m(s, p). Amiguet proves the interesting following result (see [4 (3, th. 12)]). Let m.= m.(s’ p) be as above, let a. and a¿ be two ideals of in such that and¿=m/a¿ are both of type s and of class p; then g and g' are isomorphic if and only if a. and a^ are conjugate under an automorphism of in. One consequence of this is a setting for classification problems (at given type and class): To a family of (isomorphism classes of) Lie algebras corresponds a family of ideals in m, namely a subset of a Grassmann manifold. The group Aut(m) acts on these ideals, and there is a one-to-one correspondence between orbits for this action and Lie algebras in the given family. Considerations with weight systems make it then useful (and often even sufficient) to study the orbits defined by a convenient action of a finite permutation group on some of these ideals, instead of studying the action of the "large" group Aut(m) on all of them. The examples of section 5 will hopefully illustrate the method better than any further comment. Note, however, that the "reduction" of the problem waved at above, which allows one to consider groups of permutations instead of groups of automorphisms of models, depends strongly on the fact that one looks here at algebras of maximal rank. Example 7. Let rn =rn(3, 3) be the model of type 3 (Fig. 2) and of class 3, and let e1( e2, e3 be as in example 6. Then in has the following b a s is : ej, e2, e3 which span w; e7 = [[e2, e j, e3] eg = [[e3 , e j, e2 ] e9 = [[e2, e j, e j e10 = [[e2 , e j, e2] -w h ich span C3 m-w m en = t[e3, ej, e j e12 =[[e3, e j , e3] ei3 = [[ез > e 2 e 2^ e14 = [[e3, e2],e3] The root system for in is given by Rm = B U R®m U R® in U R® m where В ={«!■ а 2, o3> 54 DE LA HARPE j>a2+2a3 i 5. VARIETIES OF NILPOTENT LIE ALGEBRAS The problem which this section illustrates is the following: Let g be a nilpotent Lie algebra of maximal rank, and let Pg be its system of weights. Classify all nilpotent Lie algebras of maximal rank having the same system of weights. As a first (trivial) example, let g be of maximal rank and suppose that Pg is equivalent to Pm(s. p); then _g is isomorphic to the model in(s, p). Section 2. 7 of Ref. [5] points out a rich class of other Lie algebras of maximal rank which are completely characterized (up to isomorphism) by their systems of weights. According to the title of the present section, we give now two examples of a different kind. Example 8. Notations being as in example 7, let R= BuR®ra U M w here M = {o?i + a2 +Q,3 > %a i +a2 }, and let P= {(e, 1) |o e R ). Let g be a nilpotent Lie algebra of maximal rank such that Pg is equivalent to P. By proposition 3, g is of type 3 and of class 3; by the digression following example 6, g is isomorphic to a quotient m/a. where m = m(3, 3) and where C3m £ a. C C2m. Comparing P and Pm. one can deduce easily that a is of the form: a= C v0 ( ma J where v is a non-zero vector in mai + P2 °3 \ « * R ~ ) (a is manufactured in order to kill those elements of Rm which do not appear in R). Now, it is sufficient to consider those automorphisms 0 of in for which ff(Tm) = Tm. where 0 is as in proposition 1 and where Tm is as in example 6. Such an automorphism permutes the basis elements ex, e2, e3 written in example 7. It follows that two ideals as a= Cv0 i and b=Cv'® ( cSФ Rm— a\ are conjugate under such an automorphism if and only V ctèf H— / IAEA-SMR-11/14 55 if v and v' are linearly dependent (otherwise said, the only possible action of the permutation group of three letters cr3 in P is the trivial action). Hence nilpotent Lie algebras of maximal rank having P as system of weights are classified by the complex projective space CP1. The last example we will give here is more typical: a family of algebras is classified by a non-trivial quotient of a Grassmann manifold (in fact,CP1 again) by a permutation group. The most typical example would be a family classified by a non-trivial quotient of a subset of a Grassmann manifold by a permutation group, but such an example would be more difficult to describe. Example 9. Let Ç be a point in the complex projective plane CP1 , and le t (zi, Z2 ) be homogeneous coordinates for Ç. In the model rn = na(3, 3) / н \ considered in example 6, let ae be the subspace (C (z1e7 - г2е8))ф ( (J) Ce¡ ) — 4 = 9 ' of rn. As aj is in C3m, it is an ideal in m and the quotient = m /a£ is a nilpotent Líe algebra. Let/t-¡ : na-* gç be the canonical projection, let x i = '/í ( e i ) f° r i e {1, .... 6} and let x^" be a non-zero vector in C3 gç , so that {хг, . . . , x7} is a basis of g? . Note that, for any two points Ç and-?7 in CP1, the two algebras ge/C3ge and gc. /С3 ge. are isomorphic. 3 Let w= 0 Cx¡ and let At(?) be the derivation of gj which extends the linear map from w to gr given by Xj '—► j q1 o th e rw ise > ^o r e a °h i = 1, 2, 3 and for each Ç eC P 1. (Check that this definition makes sense!) As A¡(f) is diagonal with respect to the basis {xx, . . ,x7}of g£ for i= 1, 2, 3, it 3 follows that Tg = 0 CAj is a maximal torus for g£ , and that g£. is of maximal rank. 1=1 L et (alta2, a 3 ) be the basis of T*jj£ which is dual to the basis (Д-j, Д2, Д3) of Tge . The system of roots can then easily be computed: R g£_ = {a1.^2>a3> a l+a2’ a 2+ a3’ a3 + a l> " 1 + a 2 + ° 3 ^ and do = dim(g? ) = 1 for each a eRgj . Hence the system of weights Pgç does not depend on Ç, and will be denoted by P onwards (Fig. 3). We will now describe a quotient space of CP1. Consider CP1 as the set of lines in the subspace E = {(vj, v2, v3) e C3/vj +v2+v3 = 0} of С 3. But then cr3 acts in standard fashion on С3 by ct3 X C 3 ------► C3 (a, (Vl, v2, v3)) '------► (v o(1) ,vd2) , v o(3) ) This action induces an action of cr3 in E, hence in CP1. The quotient space and the canonical projection will be denoted by CP1 —-— *-СР1/ст3. Alternatively, let m be a point in this quotient space and let z = t ' 1 (m), 56 DE LA HARPE FIG.3. Picture for P. where a and b are two complex numbers, each of which (but not both at the same time) may be zero. Then the orbit of z under ct3 is -i , , I a b b a + b a a + b ] . . . . t (m) = 1 — , — , ------— , ------— , ------— , ------f in general; there are Lb a a + b b a + b a J 6 three exceptional orbits which are , ~ , " lj", "j“ ~ , - 2, + 1 j- and 2 tt\ f . 2 -ïï exP v "IT)’ exp v 1 3 This being said, let be the set of all isomorphism classes of nilpotent Lie algebras of maximal rank which have P as system of weights. We have constructed a map <ÿ ' g£ It is fairly easy to check that gj is isomorphic to gç. whenever Ç and Ç' are on the same a 3~orbit; in the" other words, the map y factors as (cj3-orbit of Ç)l------► g£ Using more refined methods, G. Favre has shown that y is actually a bijection. Hence CP1/a3 represents the variety of nilpotent Lie algebras of maximal rank which admit P as system of weights. In particular, there are uncountably many isomorphism classes in <ÿ. It can be pointed out that Der(gç ) is solvable if Ç is either on a generic a3-orbit or on the 2-points orbit, but that Der(gc ) contains a subalgebra isomorphic to £>1(2, C) if Ç is on one of the two singular 3-points orbits. ACKNOWLEDGEMENT I am grateful to D. Amiguet, G. Favre and M. Favre for letting me consult some of their unpublished work and for stimulating conversations, as well as to R. Carter for the interest he showed in this work. IAEA -SM R -11/14 57 REFERENCES [1] BOURBAKI, N., Groupes et algèbres de Lie, chapitre 1. Hermann (1960). [2] DIEUDONNE, J., Eléments d'analyse 4, Gauthier-Villars (1971). See XIX, section 16, probl. 8 as well as [1], 4, ex. 18. [3] S A M E L S O N , H., Notes on Lie algebras, Van Nostrand (1969). [4] A M I G U E T , D., Extensions inessentielles d'algèbres de Lie à noyau nilpotent, Thèse, EPFL, Lausanne (1971). [5] FAVRE, G., Système de poids sur une algèbre de Lie nilpotente, Thèse, EPFL, Lausanne (1972). [6] M O S T O W , G.D., Fully réductible subgroups of algebraic groups. Amer. J. Math. 78 (1956) 200. M R 19, 1181. _ [7] FAVRE, M., Algèbres de Lie complètes, Thèse, EPFL, Lausanne (1972). [8] FAVRE, G., Une algèbre de Lie càractéristiquement nilpotente de dimension 7, C.R. Acad. Sci. Paris, Série A, 274 (1972) 1338. [9] FAVRE, M . , Algèbres de Lie completes, Thèse, EPFL, Lausanne (1972) 1533. [10J BOURBAKI, N.. Groupes et algèbres de Lie, chapitres 2 et 3. Hermann (1972). [11] S CORZA, G . , Sulle algebre pseudonulle di ordine massimo, Ann. Mat. Pura Appl. 14 (1936) 1. (I know of this paper only what is in Zbl. fur Math. 12, 102.) [12] S C H E N K M A N , E., O n the derivation algebra and the holomorph of a nilpotent algebra, M e m . Amer. Math. Soc. 1 4 (1955) 15.M R 16, 993. [13] SATO, T., The derivations of the Lie algebras, Tohoku Math. J. 23 (1971) 21. [14] DYERS, J.L., A nilpotent Lie algebra with nilpotent automorphism group, Bull. Amer. Math. Soc. 76 (1970) 52.M R 40 +2789. _ IAEA-SMR-11/15 ON INFINITE-DIMENSIONAL LIE GROUPS ACTING ON FINITE-DIMENSIONAL MANIFOLDS P. DE LA HARPE Mathematics Institute, University of Warwick, Coventry, Warks, United Kingdom Abstract ON INFINITE-DIMENSIONAL LIE GROUPS ACTING ON FINITE-DIMENSIONAL MANIFOLDS. The study of groups of diffeomorphisms has motivated a joined research by H. Omori and the author about actions of Banach-Lie groups on finite-dimensional manifolds. The present paper is essentially a list of examples which illustrate some results on these actions. 1. INTRODUCTION A rich source of problems in differential geometry is provided by variations on the following question: Question I. Let G be a topological group, let M be a manifold and ÍG XM-*M le t o: i , ■. be an action: assuming that M and p satisfy certain H |_(g,m) ^ gm properties, what can be said about G? We will use the standard terminology about group actions. In particular, the action G X M -» M is trivial — if gm = m for all geG, for all meM; effective — if gm = m for all m eM implies g = e, where e is the identity of G; transitive — if for any pair (mx . m2 } of points in M, there exists geG such that gm¡ = m2. If G is a connected Lie group (or a Banach-Lie group, see section 2 below), a transitive smooth action G X M -» M is said to be primitive — if the following holds: let mQ be some point in M and let G 0 = {geG I gm0 = m0} be its isotropy subgroup; then the Lie algebra of G0 is maximal among closed subalgebras of the Lie algebra of G. Obviously, if the action is primitive relative to some point m0 in M, then the action is primitive relative to any point in M, because we have assumed transitivity. About the notion of primitivity, see Dieudonnè [ 1, (19, problems of section 3)] and Ref. [ 2 (Introduction)]. 59 60 DE LA HARPE Before making any precision about the particular phrasing of question I which we will consider below, we give a well-known example; the first version of it is for the readers who like Lie groups, and the second for those who like Lie algebras. Example 1. Let G be a compact Lie group of dimension к acting smoothly and effectively on a smooth manifold M of dimension d. Then k-s d(d + l)/2. Proof: This consists of two inequalities, from which the above con clusion follows straightforwardly. First inequality: As the action is effective, G can be considered as a subgroup of the group of diffeomorphisms of M. As the group is compact, it is always possible to endow M with a Riemannian metric in such a way that G acts by isom etries (endow M with any Riemannian metric, and then average with respect to the Haar measure of G). Let now mQ be a point of M, let G0 be the isotropy subgroup of m 0 and let k0 be the dimension of G0. As the map J о ^ js injective by definition of the isotropy [gG 0 K Sm o subgroup at m 0, dim(G/G0) S dim(M) = d. Second inequality: Let now T be the tangent space of M at m0, which is a Euclidean space because M is a Riemannian manifold. For each Гм -» m g£G 0 let D(g): T -»■ T be the derivative at m0 of the map gm • Гм ->■ m As is an isometry, D(g) is an isometry of T, so that this defines a m ap D: pjgj where O(d) is the orthogonal group of T. As two isom etries of M are identical as soon as both their value at some point and their first derivative at this same point coincide (see Helgason [3 (I, lemma 11.2)], the map D is injective. As O(d) is of dimension d(d- l)/2, the inequality kQ S d(d - 1 )/ 2 holds. Example la. Let G be a connected simple compact Lie group of dimension к acting smoothly and non-trivially on a smooth manifold M of dimension d. Then к s d(d + l)/2. Proof: The first step consists again in proving that к - kQ S d, as in example 1. For the second step, let £ be the Lie algebra of G, let В: £ X £ -*• R be the Killing form of £ which is negative definite (because G is compact), let g^ be the Lie algebra of G0, and let m be the orthogonal complement of g^ in £ with respect to B; as В is negative definite, g is the direct sum of g^ and of m. For any Xeg^, the linear map adX j^Yh*"fx Y] orthogonal Lie algebra defined by the form В on the vector space _g; as is evidently invariant by adX, so is its IAEA-SMR-11/15 61 complement m. Let now o(m, B) be the orthogonal Lie algebra defined by the restriction of В to the space m, and let a be the kernel of the map T hen [a,jf] = [a,g0©m] = [ a, gQ ] + [a,m]ca b ec au se a is an ideal in g^ (being the kernel of a homomorphism) and also [a,m] = {0} (see the definition of a). Hence a is an ideal inj[. As G is connected and as the action is non-trivial, a cannot be the whole of g; but this implies that a = {0}, because g is simple, so that the map a is injective. Hence (k - k0)(k - k0 - 1) k0 = dim(g0) S dim(o(m, B)) = 2 which clearly completes the proof. Remarks: (1) If G is a compact semi-simple group with Lie algebra g, the Haar measure on G can be written simply in term s of the Killing form of д. so that examples 1 and la are indeed very much like each other. (2) It should be emphasized that we do need simplicity to apply the proof of example la, if that of example 1 is correct for any compact group. (3) If G is the group SO(n) acting in the standard way on the sphere then the equality к = d(d+l)/2 holds. (4) Much stronger restrictions on the dimension of G are often known, either when one restricts one's attention to a given manifold, or when one excludes a small number of particularly "symmetric" manifolds. As two out of many possible references, we quote [4] and [5] . 2. INFINITE-DIMENSIONAL LIE GROUPS Let M be a compact smooth manifold (in particular, M is finite dimensional). The set of all smooth diffeomorphisms of M is clearly a group, which we will denote by Diff(M). This group carries a differentiable structure, which has been investigated by several authors including L e slie [6], Ebin-Marsden [7], O m o ri [8]. We will not need here any deep properties of this structure, but we will only recall the two following important facts: (1) The Lie algebra С (TM) of all smooth vector fields on M plays in many respects the role of a Lie algebra for the group Diff(M). (2) Locally, Diff(M) looks like a Frèchet space. Consequently, the tra ditional tools of analysis (implicit function theorem, Frobenius theorem) cannot be applied without extreme care; indeed, these theorems hold in general in the more restricted context of Banach spaces only (see, however, several papers by Omori, both published and to appear). A Banach-Lie group is a Banach manifold which is a group and which is such that the group operations are smooth. Dealing with such groups is much easier than dealing with diffeomorphism groups, because the natural 62 DE LA HARPE context of standard analysis is that of Banach spaces (see Dieudonné [9], in particular chap.X). A short introduction to Banach-Lie groups and their Lie algebras can be found in Lazard-Tits [ 10] . Any finite dimensional Lie group is obviously an example of a Banach- Lie group; others can easily be given as follows. Example 2. Let H be a separable real or complex Hilbert space and let L(H) be the algebra of all bounded linear operators on H. Then the group GL(H) of all invertible elements in L(H) is a Banach-Lie group, with the manifold structure being given by the open inclusion of GL(H) in L(H). The space L(H) furnished with the product f L(H) X L(H) — L(H) J is denoted by gl(H); it is a Lie algebra and [ (X, Y) и- [X, Y] = XY - YX a Banach space, and the multiplication is continuous; ||[X,Y]|| S c || X |¡ Y || ; in this last inequality, с is a constant which can be taken equal to 2. An object such as gl(H) is called a Banach-Lie algebra. The exponential map of the Banach-Lie group GL(H), which is defined exactly as the exponential map of any finite dimensional Lie group, is given by the familiar power series '¿.(H) - GL(H) XH exp X = ^ ¿ X n n = 0 Example 3. Suppose first that H is a finite dimensional complex Hilbert space, so that GL(H) is denoted,as usually, by GL(n,^). The following groups can be defined as subgroups of GL(n, <£), in a simple way and by algebraic equations; U(n), U(p, q) with p + q = n, U*(n) when n is even, GL(n,R). (Notations as in Helgason [3 (IX, 4)].) Suppose now that H is an infinite-dimensional complex Hilbert space. Similar constructions can be carried over to this case; for example, the unitary group U(H) of H is defined by (VeGL(H) | VV* = V*V = 1}; it is easy to check that U(H) is a Banach-Lie group with Lie algebra u(H) = {Xegl(H) I X* = -X}. A systematic study of such Banach-Lie groups has been started, se e R ef. [ 11 ]. A second approximation to the question of interest here can now be formulated: Question II. Let M be a finite dimensional manifold; is it possible to find an infinite dimensional Banach-Lie group G and an action of G on M satisfying certain preimposed conditions? Any effective action of a group G on M defines obviously an injection of G in Diff(M). Hence yet another phrasing: IAEA-SMR-11/15 63 Question lia. If M is as above, what are the subgroups G of Diff(M) which can be furnished with a Banach-Lie group structure such that the action G X M -» M induced by the evaluation map Diff(M) XM-*Mis sm ooth? About the relationship between this question and other considerations on group Diff(M), remember Professor Eells' lecture in this Summer College ("The diffeomorphism group in analysis"), which is not published in these Proceedings. 3. BANACH-LIE GROUPS WHICH DO NOT ACT ON FINITE-DIMENSIONAL MANIFOLDS Let G be a real Banach-Lie group acting smoothly on a manifold M of finite dimension d. Let m0 be a point of M, and let G0, £ and gQ be defined as in section 1. Choose an element X £ |, and let IR G J be the corresponding one-parameter subgroup of G. For jtH- exp(tX) d each point m in M, the quantity ^ ((exp(tX))m) is a tangent vector at t = o m to M; hence one has a vector field on M which will be denoted by ти- X(m). Everything being smooth, this vector field has a Taylor series X a t m 0. L e t now & be the space of "formal power series of local vector fields at the origin of IRd", namely the space of all expressions of the form Э A„ II Эх1 i = l a £ N a where the A^'s are real constants and where we have used the traditional notation for multi-indices. The space JM s made a real Lie algebra in the obvious way, and the construction above defines a homomorphism g - S * - Ф - Хи- X F o r each k e { -1 , 0, 1, 2, ...}, define the subspace d Jv = 1 p e , , , ylp'Ii = l a 6 N d °,s" lalak+l 64 DE LA HARPE Then the family (Jk)ke {-i, o, ..} *s a filtration of the Lie algebra â*. This means that (1) J-i 3 J0 э Jx э ... d Jk d Jk+1 э ... with Jk = & and Jk = {0} к = -1 k = -l (2) [Jk,J(] CJkt(for all k,ie{-l, 0, 1, ...} (3) dim(Jk/Jk + 1 ) < oo for all ке{-1, 0, 1, ...} Moreover, the resulting filtered Lie algebra is transitive: (4) for every ke{0, 1, 2, ...}, for every P eJk such that P ф Jk+i • there exists XeS*" such that [X, P] ф Jk (That condition 4 is a transitivity requirement is best seen by looking at Dieudonnê [ 1 (problem 5 d of section 19.3)]. Defining now gj^ = Ф 1(Jk) for all k e{-l, 0, 1, . . .} , we obtain a filtration (ёк)ке{-1 o } °^ Banach-Lie algebra g_. (See remark just before example 5 for the condition O i i = i°b) к =-1 The study of (possibly infinite dimensional) filtered Lie algebras, initiated by E. Cartan [ 12], has produced a very rich and deep theory; as an elementary applicationof it and of classification results [ 16], the following two theorems have been proved in Ref. [ 13] : Theorem 1. Let G be a connected Banach-Lie group, which is second- countable, and which acts smoothly and effectively on a finite dimensional manifold M. If the action is primitive, then G must be finite dimensional. Theorem 2. Let g be the Lie algebra of an infinite dimensional Banach-Lie group G. Suppose g has no proper closed finite codimensional ideal. Thenjf has no proper closed finite codimensional subalgebra. In particular, any smooth action of G on a finite-dimensional manifold is trivial. The detailed proofs cannot be given here. Their principle is, however, quite simple, and is best exposed by the following. Example 4. Let h be the Lie algebra of vector fields on R generated by the elements Э Э 2 _Э_ n Э Эх ’ X Эх ' X Эх ' ‘ ‘ ’ X Эх ’ Then there is no Banach-Lie algebra £ such that h is (isomorphic to) a subalgebra of g. lAEA-SMR-11/15 65 Proof: Suppose indeed that there exists a norm || || on h and a constant с such that || [ A, B] || é с || A j| || В || for all A, B e h. Then U П U S с X Эх ’ X Эх Эх for all integer n, which is clearly absurd. Hence the algebra Ф(£) (notations as before theorem I) cannot be isomorphic to the algebra h of example 4, because 'í'(g) is a Banach-Lie algebra. Using the known classification of real "primitive filtered Lie algebras", one can exclude all infinite dimensional cases by similar computations; hence theorem I and, with hardly any more work, theorem II. Remark: In supposing above that Ф(|[) is a Banach-Lie algebra, we have implicitly assumed that the map Ф is injective; this is always the case if the action of G on M is analytic; though theorems I and II are true for smooth actions in general, we cannot enter into details here and we refer to Ref. [ 13]. Example 5. Let GL(H) be as in example 2, with H a real infinite dimensional Hilbert space. Then any action of GL(H) on a finite dimensional manifold is trivial. Proof: The ideal structure of the associative algebra L(H) is well- known (see for example Schatten [ 14 (I. 6)] . Using results on the Lie structure of an associative algebra as in Bourbaki [ 15 (1, ex. 7)], one can easily check that any proper ideal of ¿1(H) Is contained in the ideal X a real number Y a compact operator on H so that any proper ideal of gl(H) is a fortiori of infinite codimension. Hence theorem II applies and ends the proof. The conclusion could naturally be expected after example la. 4. A BANACH-LIE GROUP WHICH ACTS EFFECTIVELY AND TRANSITIVELY ON R2 The hypothesis of primitivity is essential in theorem I, as will be shown in this section. Let E be the space of those analytic real functions of one real variable whose Taylor series at the origin is of the form sup I annl I Lemma 1. For each s e IR and feE, fs is in E; moreover, the ftR X E - E mapping r):4 is smooth. [(s,f)H- fs Sketch of proof: oo 00 n Step one: If f(x) anx n, then fs(x) xksn_k = n = 0 n = 0 k = 0 00 00 a" (n-k)i ki) xk- ||f||e'sl =Ik=0 Œ n=k Hence iifsii- k€Nsup(I n=k |s,n’k)s which proves the first assertion. Step two: For each integer m, let f(m> denote the m - th derivative of f. It is trivial to check that f^m' e E and that || f(m^ || s || f || . For each keN and for each selR, let now К f(x + s) - ^ - ¿ 7 f (m) (x)sm it is obvious that (pk(f, s) belongs to E; direct computations using Taylor's formula show, moreover, that II к k-1 xK (f + g)(x + s+ t) - j T ^ f(m) (x + s)tm + ^ g (m,(x + s)tm| m=0 m=0 2 и 2 and let 6 be the number J |t| + ||g|| . Then, if 6 f 0, k ¿ Fk (f,g; s,t) (x) j f t x + s + t) f k-1 + 7E 1g(x+s + t) g(m) (X + S) T IAEA-SMR-11/15 67 so that 11^-Fk (f,g; s,t) || S p p r I K (fs, t) || + || s |t| e |s| +lt| ||f|| + e |s| +lt| Il g II Hence lim Ц-i Fk (f, g; s,t)|| = 0. 6 -> о Step three: For each keN and assuming it exists, the к-th derivative of the map r¡ at (s, f) is denoted by (Dkrj) (s,f) and is a symmetric multilinear map (R ® E) X ...... X (R ® E) - E к tim es We claim that this derivative exists and is given as follows: (Dkr)) (ti> gj )(t2> g2) • • • (tk. gk) is the function which takes at x the value к f(k) (x+sH-lV ,tk + ^ g f 4 (x + s H j . . tj. . t k j=l In order to prove this claim, it is sufficient to prove that (Dkrj) (t, g) .... (t, g) is the function which takes at x the value Let now G be the Banach-Lie group whose underlying manifold is the Banach space R Ф E and whose product is defined by (s,f)(t, g) = (s + t, ft +g). Lemma 1 shows that the group operations are smooth, so that G is indeed a Banach-Lie group. ÍEXR-R Lemma 2. Let f be the evaluation map -j . Then f is smooth. (f, x) и- f(x) k Proof. For each keN, it is easy to check that (D f) is the symmetric multilinear map defined by ^ (° kç)(f,x) t e r M b . y2 >■ • • tek • yk) = f(k) (*&! y2 • ■ • yk к + Z i"4 (Х)У1 "•У! •••Ук Hence the lemma. j = i 2 Let now G be acting on R by G X R2 R2 ( . Lemma 2 shows that this action is smooth. ((s, f), (x, у)) и. (s +x, f(x) +y) 68 DE LA HARPE It is trivial to check that this action is effective and transitive. The 2 i isotropy subgroup of G at the origin of R is G0 = {(s, f)£G| s = 0 and f(0) = 0}. In agreement with theorem I, this action is not primitive; indeed: GjCEcG, the codimension of G0 in E is +1 and the codimension of E in G is +1. (E is naturally identified with the subgroup {(s,f)eG| s = 0} of G . ) Geometrically speaking, it should be noted that the action described in this section has a certain "symmetry" (this is not a technically well- defined term, at least not at present) which can be pointed out as follows. Let (s, f) be a fixed element of G; at each point m of R2 , let v(m) be the vector (s, f)m - m; then the vector field v on R2 is invariant by vertical translations. It is instructive to draw pictures, say when the element of G is (0, f) or (i, f) and when f is the function defined by f(x) = x. 5. CONCLUDING REMARKS The moral to retain of theorem I is that the study of Banach-Lie groups and that of groups of diffeomorphisms are in fact very different subjects of investigations, even though both are typically infinite dimensional. The example of section 4 suggests that actions of (possibly infinite dimensional) Banach-Lie groups on finite dimensional manifolds arise in connection with families of vector fields posessing certain "symmetries". REFERENCES [1 ] DIEUDONNE, J . , Elements d'analyse 4, Gauthier-Villars (1971). [2] GUILLEM1N, V., Infinite dimensional primitive Lie algebras, J. Differential Geometry 4 (1970) 257. MR 42 +3132. [3 ] HELGASON, S . , Differential Geometry and Symmetric Spaces, A cadem ic Press (1962). [4] HSIANG, Wu-Yi, On the bound of the dimensions of isometry groups of all possible riemannian metrics on an exotic sphere, Ann. of Math. 8^(1967) 351. MR 35 +4935. [5 ] KOBAYASHI, S ., Transformation Groups in D ifferential Geometry, Springer (1972). [ 6] LESLIE, J. A ., On a differential structure of the group of diffeomorphisms, Topology 6 (1967) 263. MR 35 +1041. [7] EBIN, D. G ., M ARSDEN, j., Group of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. 92 (1970) 102. MR 42 +6865. [ 8] OMORI, H ., Local structures of groups of diffeomorphisms, J. M ath. Soc. Japan 24 (1972) 60. [9 ] DIEUDONNE, J . , Les fondements de l'analyse moderne, Gauthier-Villars (1963) or A cadem ic Press (1960). [10] LAZARD, M ., TITS, J., Domaines d'injectivité de l’application exponentielle, Topology 4 (1965/66) 315. MR 32 +3518. [11] DE LA HARPE, P ., Classical Banach-Lie algebras and Banach-Lie groups of operators in Hilbert space. (To appear as Mathematics Lecture Notes 285, Springer.) [12] CARTAN, E., Ouvres complètes, (Part II, vol.2) Gauthier-Villars (1953). [ 13] OMORI, H ., DE LA HARPE, P ., About interaction between Banach-Lie groups and finite dimensional manifolds, J. Math. Kyoto Univ. 12 (1972) 573. MR£7_. 4286. [14] SCHATTEN, R., Norm Ideals of Completely Continuous Operators. Springer (1960). [15] BOURBAKI, N., Groupes et algèbres de Lie, chapitre I, Hermann (1960). IAEA-SMR-11/15 ADDITIONAL BIBLIOGRAPHY From the vast literature concerned with the classification results used to prove theorems I and II, one can quote besides Refs [ 2] and [ 12] the four following: KOBAYASHI, S ., NAGANO, T . , On filtered Lie algebras and geom etric structures I-V , J. M ath. Mech. (1964-1966). MR29 + 5961, 32 +2512 and 5803, 33 +4188 and 4189. SINGER, I .M ., STERNBERG, S . , The infinite groups of Lie and Cartan, part I (the transitive groups), J. d'an. m ath. 15 (1965) 1. MR 36+911. MOR1MOTO, T ., TANAKA, N., The classification of the real primitive infinite Lie algebras, J. Math. Kyoto University 10 (1970) 207. MR 42 +3133. SCHNIDER, S ., The classification of real prim itive infinite Lie algebras, J. Diff. Geom. 4 (1970) 81. On the notion of a Banach-Lie group as introduced in section 2, see the recently published Groupes et algèbres de Lie, chap. 3, by Bourbaki (Hermann, 1972). IAEA-SMR-11/16 SOME PROPERTIES OF INFINITE DIMENSIONAL ORTHOGONAL GROUPS P. DE LA HARPE Mathematics Institute, University of Warwick, Coventry, Warks, United Kingdom Abstract SOME PROPERTIES OF INFINITE-DIMENSIONAL ORTHOGONAL GROUPS. The standard notion of a (real or complex) Lie group can be extended in many ways. One of them leads to the study of the so-called Banach-Lie groups and of their Lie algebras. An important part of the theory of finite-dimensional Lie groups can easily be seen to carry over to the Banach case; but several interesting properties characteristic of infinite dimensions have recently been (or are now being) discovered. One of them is discussed in this paper. Section 1 recalls properties of classes of compact operators acting on a Hilbert space H (the von Neumann-Schatten ideals). Section 2 contains the definitions of the groups of interest here, which are groups of orthogonal operators on H. Section 3 shows how to compute the homotopy types of these groups; section 4 very briefly indicates a connection with the theory of representations of the canonical anticommutation relations of quantum physics. The idea of section 3 is due to H. Pittie. 1. REVIEW OF OPERATOR THEORY (VON NEUMANN-SCHATTEN IDEALS) Let H be a real Hilbert space of an infinite number of dimensions. The Banach algebra of all continuous linear operators on H will be denoted by L(H), and that of all compact operators on H by C(H). If X is in C(H), it is classical that the spectrum of X (by which we mean the usual spectrum of the operator XŒ which extends X linearly on the complexification HŒ of H) contains at most countably many points, which are all isolated eigenvalues (of Xa-) with the only possible exception of 0. The operator XX* (where X* is the adjoint of X) has a unique positive square root, which is again a compact operator, denoted by [X]. The spectrum of [X] consists of, at most, countably many isolated strictly positive numbers and of 0. We shall denote by (nDQi))n = i,2,... an enumeration of the eigenvalues of [X], repeated according to their multiplicity, and ordered such that Ml(X) È M2(X) è ...... ё м„(Х) ё ...... If the spectrum of [X] is finite, the numbers /jn(X)'s are all 0 for n large enough. In any case, the sequence (/un(X)) tends towards zero. In all of what follows, p is either a real number larger than one, or the symbol + oo : р еИ , 1 È p g + oo 71 72 DE LA HARPE Let us now define the von Neumann-Schatten ideals Cp(H): Пn = 1 furnished with the norm || ||p defined by 1/p llxllp = (V(Mn(X))],p) Ifp=co: C«(H) = C(H) furnished with the norm || || (or || || „) induced from the Banach algebra L(H). We recall, in a first proposition, some properties of these: Proposition 1 (i) Cp(H) is an ideal in L(H). (ii) Cp(H) is a real involutive Banach algebra. (iii) If X is in Cp(H) and if U, V are orthogonal operators on H, then IIUXV ||p = IIX ||p . (iv) If X is in Cp(H) and if A, В are in L(H), then ||a X b |p S||a|| ||х||р||в||. (v) If 1 sp sp 's », then the injection Cp(H)->- Cp'(H) is continuous. (vi) Let (en)j=i 2 be an orthonormal basis of H; for each n, let Mn be the set of those operators which map the span of {eb . . . , en} into itself and its orthogonal complement onto zero; let M„ be the union of all the Mn 's. Then M„ is dense in Cp(H); in particular, the set of finite-rank operators on H is dense in Cp(H). Detailed proofs are given, e. g. by Schatten [12]. The most interesting cases in analysis are p = l (nuclear, or trace-class, operators), p = 2 (Hilbert-Schmidt operators) and p = °o (compact operators); typical examples can also be found in Ref.[2], section 15.4, problem 14, and section 15.11, problem 7. 2. GENERAL LINEAR GROUPS AND ORTHOGONAL GROUPS OF OPERATORS ON H The space L(H) can be viewed as a (real) Banach-Lie algebra, with the product defined by [X,Y] =XY-YX, and will in this case be denoted by _gl(H). Then j[l(H) is the Lie algebra of the Banach-Lie group GL(H) of all invertible operators on H. Some more details are given in the second part of section 2 of Ref. [6]. Let o(H) be the sub Banach-Lie algebra of all skew-adjoint operators in gl(H). It is easy to check that £(H) is the Lie algebra of the Banach-Lie group O(H) consisting of all orthogonal operators on H. Similarly, the space Cp(H) can be viewed as a Banach-Lie algebra which will be denoted by gl(H; Cp). Let now GL(H; Cp) = {T eG L (H )|T -IeC p(H)} IAEA-SMR-11/16 73 where I is the identity operator on H. It is a group (indeed, a normal subgroup of GL(H)) because Cp(H) is an ideal in L(H). The map i G L ( H ; C p ) ------C p ( H ) [ T I------T - I is a bisection of GL(H; Cp) onto the open subset of Cp(H) containing those operators for which +1 is not an eigenvalue. This endows GL(H; Cp ) with the structure of a Banach manifold. Finally, the group operations are smooth with respect to this structure, because the multiplication in Cp(H) is continuous (hence analytic; see Ref. [1], example 8.12.9). In other words GL(H; Cp) is a Banach-Lie group; it is very easy to check that gl(H; Cp) is its Lie algebra and that the exponential map is given by the traditional power series. Finally, let o(H; Cp) be the Banach-Lie algebra of all skew-adjoint operators in gl(H; Cp). This is the Lie algebra of the Banach-Lie group 0(H; Cp) = (TeO(H) |Т - ieC p(H)} = {T6GL(H; Ср)|тт* = T*T = 1} It is an easy and instructive exercise to translate for G1(H; Cp) and 0(H; Cp), section 7.2 of Lang [8], who considers G1(HŒ) and U(HŒ ), their exponential maps and the polar decomposition of invertible operators. Note that the function j °^H' c p)x ^ H , Cp)^ >- R+^ ^ defines a bi-invariant distance on the group 0(H; Cp); the associated topology is that underlying the manifold structure of 0(H; Cp) defined above. Our target is to study the homotopy type of these topological groups, in terms of the homotopy types of the (finite-dimensional) classical Lie groups O(n). 3. HOMOTOPY TYPES It follows trivially from polar decomposition (see Lang [8], chapter 7, proposition 6) that G1(H) and O(H) have the same homotopy type, and the same holds for GL(H; Cp ) and 0(H; Cp). Hence, it is sufficient to consider orthogonal groups only in this section. The most famous result is Kuiper's theorem [7]: Proposition 2. The Banach-Lie group O(H) is contractible. The proof of proposition 2, or even a sketch, is beyond the scope of this lecture. The homotopy type of the groups 0(H; Cp) is much easier to compute and has been worked out (essentially independently) by Elworthy, Geba, Palais, Svarc, Tromba (see the references given in Ref. [4]). The method of proof of proposition 3 below has,to our knowledge, not yet been published. We have learned it from H. Pittie. 74 DE LA HARPE Let (en)n = ii2l...be an orthonorm al basis of H. For each n, we identify the Euclidean space Rn to the span of {e j, .. ., en} in H and the compact Lie group O(n) to the subgroup of 0(H; Cp) consisting of those operators which map Rn onto itself and whose restriction to the orthogonal complement of Rn in H is the identity (O(n) = 0(H; Cp)DMn , where Mn is as in proposition l(vi)). These identifications give rise to the standard inclusions 0(n)C 0(n +1 )C . ..C O (co)----}-----► 0(H; Cp) r 00 where O(°o) is the topological group O(n), namely the group U O(n) furnished with the inductive limit topology. n_1 Proposition 3. The inclusion O(oo)----— >-0(H; C p) is a homotopy equivalence. Proposition 3 reduces the computation of the homotopy type of 0(H; Cp) to that of the stable orthogonal group, which is known (see, e.g. Milnor [10], part 4). F irst, we prove three lem m as: Lemma 1. Let T0 and Tj be two elements in 0(H; Cp ) such that ||Tj - T0||p Proof. By proposition 1(iii), H t'^Tj-Tq)! s || T As an immediate corollary, if X is a topological space and if f0, fj are two continuous maps from X to 0(H; Cp ) such that sup ||fi(x) - f0(x)|| < 1, then f0 and f j are homotopically equivalent. xeX Lemma 2. Let X be a compact polyhedron and let f: X — ► 0(H; Cp) be a continuous map. Then there exists an integer m and a map g: X ----►O(m) such that f and g are homotopically equivalent in 0(H; Cp). Proof. For each integer n, consider O(n) with its standard Riemannian structure (see, e. g. Milnor [10], part 4). Let e be a strictly positive real number, smaller than 1/5, and smaU enough for all the balls of radius 3 e in O(n) to be convex. (This means that, given any two points inside such a ball, there exists a unique geodesic segment inside this ball which goes from one point to the other; it is a fact that the intersection of two of these balls is again convex; such an e can be chosen independently of n. ) Let d be a distance on X compatible with the topology. As X is compact, f is uniformly continuous and there exists a strictly positive real number e' such that, for all x, y G X with d(x, y) < e1, the inequality || f(x) - f(y ) ||p < e holds. Let К be a triangulation of X such that each simplex о of К is con tained inside an open ball of radius e1 in X; the vertices of К will be denoted by Xj, . . . , x k. As O(oo) is dense in 0(H; Cp) by proposition l(vi), for each ie{l, . . . , k} there exists an integer m¡ such that the distance between f(x¡ ) and 0(m;) is smaller than e. Let m be the largest of the m¡'s, Then, for any x eX , the distance between f(x) and O(m) is sm aller than 3 e. IAEA-SMR-11/16 75 For each ie { l, . . . , k}, let y¡ be a point in O(m) such that || f(x¡) - y¡ ||p < e. Define a map g: X ----► 0(H; Cp) as follows: On the vertices of K, put g(x¡) = y¡. Let now a be a closed simplex in К which is contained in no other closed simplex than itself (one of the "largest" simplices); if xj, . . . , Xj(0j are the vertices of a, then î(xi), . . . , f(xj(c)) are contained in some convex ball, say B0, of radius e in O(m); hence, yi , . . . , yj(0) are contained in some convex ball BJ, of radius 3 e in O(m); extend g to map continuously a in B¿,, with the following condition: if some subsimplex of ó is also a subsimplex of another of the "largest" simplices, say f, then the corres ponding yj's must also be contained in the corresponding ball В| of radius 3 e; (there is no obstruction to this extension of g because all the convex balls and their intersections are contractible). This procedure can be repeated for all the closed simplices of К as above, one after the other, taking each time care of those subsimplices where the map may already have been defined. One obtains, in this way, a continuous map g: X ----►O(m) such that sup j|f(x) - g(x)|| < 5 e. Hence, by lem m a 1, f and g are homo- x e X topically equivalent. Lemma 3. Let X be a compact polyhedron and let f0, fj : X-----»-0(oo) be two continuous maps which are homotopically equivalent in 0(H; Cp). Then fQ and fj are homotopically equivalent in O(oo). Idea of proof. Let F: X x [0,1] ----»-0(H; Cp ) be a homotopy of f0 towards fj in 0(H; Cp). Let L be an ad hoc triangulation of Xx[0,1] which contains a subtriangulation К of (X x{0})U (Xx{l}). By hypothesis, F maps К in the subgroup O(oo) of 0(H; Cp). Proceeding very much like in the proof of lemma 2, it is possible to define a map G: Xx [0, 1 ] ----►O(oo) which is close to F, and which actually coincides with F (namely, with f0 and fj) on IК I =(Xx{0})u (Xx{l}). It follows that f0 and fj are homotopically equivalent in O(oo). Proof of proposition 3. The map j induces a group homomorphism j¡ : nt(0(oo))----»• rij (0(H; Cp )) for all ie { l,2 ,. . .} and a set homomorphism between the n0's. These homomorphisms are surjective by lemma 2 and injective by lemma 3. Hence, all the maps j¡'s are isomorphisms, which is expressed by saying that j is a "weak homotopy equivalence". A standard technical trick allows us to conclude: The inductive limit O(oo) has the homotopy type of an ANR (Hansen [3], corollary 6. 4) and the Banach mani fold 0(H; Cp) is an A.NR. Hence Whitehead's lemma applies (see, e. g. Palais [11], section 6.6) and j is a homotopy equivalence. 4. SPIN GROUPS AND CAR As a corollary to proposition 3, the group 0(H; Cp ) has two connected components (also given by the sign of the determinant in the case p = 1); and its connected component 0+(H; Cp ) has Z2 as fundamental group. It follows from general principles that the universal covering Spin(H; Cp) of 0 +(H; Cp) is again a Banach-Lie group, with the same Lie algebra as that of 0(H; Cp). To construct this two-fold covering explicitly (at least, 76 DE LA HARPE in the easiest case, i. e. p = 1), it is possible to proceed as follows: The x j |x |2 can be made a real involutive normed algebra, with the norm of XeCl(H) being defined as |x|„ = sup {X€ IR IXX* - X2 is not invertible} and with the involution X ----►X* of C1(H) being the (unique) anti-automorphism of C1(H) whose restriction to H is the identity (H is canonically identified with a subspace of C1(H), which is then a system of generators of this algebra). The com pletion Cl^H) of C1(H) with respect to | |„ is a real C*-algebra. By the universal property of Clifford algebras, any orthogonal operator UcO(H) can be extended to a *-automorphism of Cli(H) which we will denote by C1(U). Consider now a "''-automorphism of Cli(H) which maps H onto itself (one of the so-called "Bogoliubov automorphisms"), so that it can be written as C1(U) for some UGO(H). Then C1(U) is said to be inner if there exists an invertible element ueCli(H) such that C1(U)(X) =иХи_1 for all XeCli(H); an inner automorphism C1(U) is said to be even if u can be chosen in Cl+i(H) [where Cl^H) =C1Î(H) © Clj(H) is the canonical Z2-graduation of the Clifford algebra Cli(H)]. A basic result about these inner automorphisms is due to Shale and Stinespring [14]: Proposition 4. If U e O ( H ) , then C1(U) is inner and even if and only if U e O +(H; Ci). One of the interests of proposition 4 is its interpretation in terms of the canonical anticommutation relations (CAR) of quantum physics (see also Shale and Stinespring [13] and Slawny [15]). As a consequence of this, it is possible to prove: Proposition 5. Let Spin(H; Cj) be the group of those invertible elements u£ Cl^fH) which are unitary [namely uu* =u*u = 1], even [namely uECl^H)] and such that uHu"1 = H. Then Spin(H; Ci) can be endowed with the structure of a Banach-Lie group such that the map Spin(H; C i ) ------*- 0 +(H; Ci) H----►H \ X I— »■ uxu"1/ is the universal covering of 0 +(H; Ci ). That the range of the map py is precisely 0 +(H; Ci ) is given by proposition 4. The description of the Banach-Lie group structure on Spin(H; Ci) and the proof of proposition 5 are given in Ref. [5]. The explicit constructions of the coverings Spin (H; Cp ) ----»-0+(H; Cp ) (when p > l) will hopefully be worked out in the near future. REFERENCES [1] DIEU DONNÉ, J. t Fondements de l ’analyse moderne, Gauthier-Villars (1963). [2] DIEUDONNE, J . , Eléments d ’analyse, 2. G authier-Villars (1968). [3] HANSEN, V. L., Some theorems on direct limits of expanding sequences of manifolds, Math. Scand. 29 (1971) 5. IAEA-SMR-11/16 77 [4] DE LA HARPE, P ., Classical Banach-Lie algebras and Banach-Lie groups of Operators in Hilbert space, Springer Lecture Notes in Math. 285 (1972). [5] DE LA HARPE, P ., The Clifford algebra and the spinor group of a Hilbert space (to be published in Compositio Mathematica). [6] DE LA HARPE, P ., On infinite-dimensional Lie groups acting on finite-dimensional manifolds (these Proceedings). [7] KUIPER, N., The homotopy type of the unitary group of Hilbert space, Topology 3 (1965) 19. [8] LANG, S., Introduction aux variétés différentiables, Dunod(1967). [9] LAZARD, М ., Groupes différentiables. "Neuvième lepon" of a course to be published in book form. [10] MILNOR, J . , Morse Theory, Princeton University Press (1963). [11] PALAIS, R. S., Homotopy theory of infinite-dimensional manifolds, Topology 5 (1966) 1. [12] SCHATTEN, R., Norm Ideals of Completely Continuous Operators, Springer (1960). [13] SHALE, D ., STENESPRING, W .F ., States of the Clifford algebra, Ann. Math. 80 (1964) 365. [14] SHALE, D ., STINESPRING, W. F ., Spinor representations of infinite orthogonal groups, J. Math. Mech. 14 (1965) 315. [15] SLAWNY, J., Representations of canonical anticommutation relations and implementability of canonical transformations, Thesis, The Weizmann Institute of Science, Rehovot (1969); see also Commun. Math. Phys. 22 (1971) 104 (same title). IAEA-SMR-11/17 THEORY OF RESIDUES IN SEVERAL VARIABLES P. DOLBEAULT Department of Mathematics, University of Poitiers, Poitiers, France Abstract THEORY OF RESIDUES IN SEVERAL VARIABLES. The standard notion of a residue at a pole of a meromorphic 1-form on a surface extends to arbitrary forms on analytic manifolds, and can be expressed on more general spaces in (co)homological terms. The paper recalls the definition of the residue homomorphisms, reviews various contributions due to Leray, Norguet, the author and many others and, finally suggests several topics and problems for further studies. 0. INTRODUCTION 0.1 Let X be a Riemannian surface and let u be a m erom orphic differential form of degree 1 on X; in the neighbourhood of a point where z is a local co-ordinate, we have w = f(z)dz where f is a meromorphic function. Let Y = {aj}jeI be the set of poles of u; for every j G I, let Yj be a positively oriented circle with centre a¡ such that the closed disk of which is the boundary does not meet Y in any point different from a¡; let Resa.(u) be the Cauchy residue of и at a^-, then for any finite subset J of I and for1 any family (nj)jeJ of integers, or real or complex numbers, we know the formula of residues (1) 0.2. Moreover, if X is compact the condition (2) is a necessary and sufficient condition for a given set of numbers (Resa. )jeI to be the residues of a meromorphic form with poles at the points aj. Poincare was the first to give a convenient generalization of the notion of residues for closed meromorphic differential forms in several complex variables (1887) [27]. 79 80 DOLBEAULT 0. 3. Theory of Leray-Norguet Starting from Poincaré's work, Leray (1959) [18] , then Norguet (1959) [22,23] generalized the situation of sub-section 0. 1. to the case of a complex analytic manifold X of any finite dimension, Y being a complex sub-manifold of X of codimension 1; u a closed differential form of degree p on X, of class С outside Y and having singularities of the following type on Y: every point x 6 Y has a neighbourhood U in X over which a complex local co-ordinate function s is defined such that YH U= {yG U; s(y) = 0} ; then и I U is equal to a /s k where a is C" on U and к 6 N. More generally, a semi- meromorphic differential form on X has a local expression a/f where a is a C” differential form and where f is a holomorphic function. (a) The topological situation studied by Leray and generalized by Norguet is the following: Let X be a topological space and Y be a closed subset of X, we have the following exact cohomology sequence with complex coefficients and compact supports ■ • • - H^(X) - HcP(Y) - HcP+1(X \Y ) - HP+1(X) - • • • (3) If Y and X\Y are orientable topological manifolds of dimension m and n respectively, the duality isomorphism of Poincaré defines, from 6 V, the homomorphism 6 : H° (Y )-H ° n(X\Y) m-p ' ' n-p-1 ' Hence, from the universal coefficient theorem, the residue homomorphism r = l6 : Hn"p’1{X\Y) Hm"p(Y) and the formula of residues ■(ôh, c)> = in which h G H ^.p(Y) and c G Hn"p"1(X\Y). Clearly, formula (4) generalizes formula (1) where с is the cohomology class defined by ш | X\Y and h is the homology class I Vi jej (b) Suppose now that X is a complex analytic manifold. Given a semi- meromorphic form w on X, C" except on a submanifold Y of codimension 1, IAEA-SMR-11/П 81 such that, locally, и = a/s where s = 0 is a m inim al local equation of Y, we say that w admits Y as a polar set with multiplicity 1. Then, locally, и = (ds/s) A(//+ d0 where Ф and в are С” ; Ф | Y has a global definition and is called the residue form of и and its cohomology class in Y is the image, by r, of the cohomology class defined by u |x \Y on X\Y. So, in this particular case, we have an interpretation of the residue homomorphism in terms of differential forms. (c) Theorem of Leray ([18] , theorem 1, p. 88): Let X be a complex analytic manifold and Y be a submanifold of X of codimension 1, then every class of cohomology of X\Y contains the restriction, to X\Y, of a closed semi-meromorphic differential form having Y as a polar set with multiplicity 1. When dimŒ = 1, this theorem reduces to the statement in sub-section 0.2. (d) Composed residues: Let Yj, . . ., Yq be complex submanifolds of X of codimension 1, in general position and let y = YjH ... П Yq, then by com position of the residue homorphisms HP(X\YjU . . . UYq) - H5'1 (Yj\ Y2U. . . U Yq) - HP'2(Yin Y2 \ Y3U . . . U Yq ) . . . - HP 4(y) we obtain the composed residue; after a permutation of the Y^, the composed residue is multiplied by the signature of the permutation. 0.4. Generalizations. From 1968, generalizations of the definition (a) of the residue homomorphism have been used for spaces more general than complex manifolds; (b) semi-meromorphic differential forms with polar set having arbitrary singularities and generalizations of such forms have been used to give interpretations of the residue homomorphism; the theorem of Leray (c) and composed residues (d) have been generalized; finally, attempts to get residue formulae have been made [34, 26]. We shall review these results, except those concerning residue formulae. In fact, a small part of this program has been realized before Leray's work (1951-57) [16,17, 33, 2, 5]. But for a more complete realization, the following newer techniques have been used: Borel-Moore homology ([4] , chapter 5); local cohomology (Grothendieck [13] ); Hironaka's local resolution of singularities of complex analytic spaces [15]; properties of semi-analytic sets (fcojasiewicz [19] , Herrera [9] ). This paper does not contain the review of recent work on the subject by Norguet (residues and q-convexity [25] ), by Gordon on a geometrical theory of residues [12] , by Griffiths on residues on algebraic manifolds and by specialists in algebraic geometry (Grothendieck, Hartshorne, . . . ). Let us also remark that residues can be studied on infinite-dimensional spaces [1]. 1. DEFINITION OF THE RESIDUE HOMOMORPHISM Homology and cohomology groups are supposed to be with coefficients in(D. 82 DOLBEAULT 1. 1. Homological residue [8a] Let X be a locally compact, paracompact space of finite dimension and Y be a closed subset of X. In the Borel-M oore homology (homology of locally compact spaces) [4] , we have the following exact homology sequence: • • • ~ H q+l( X ) ~ H q+1 ( X 4 Y ) 6* H q(Y) ^ H q(X) ^ ‘ ‘ ^ It is transposed from the exact cohomology sequence (3). When X\Y and Y are orientable topological manifolds of dimension n and m, respectively, we have the following diagram: P x * P +i Hc (Y) A*. Hc (X\Y) H m'p(Y) J -X _ Hp(Y) r I (*) x(Q) n-p-l n 6 = l(5 H (X\Y) x \ Y * H (X\Y) •*------p+i where pY and px\Y are ^ e duality isomorphisms of Poincaré, where (*) is anticommutative and where (Q) gives a residue formula; 6* will be called homological residue homomorphism and gives a generalization of r. 1.2. Cohomological residue For any topological space X and any closed subset Y of X, we have the exact sequence of local cohomology ------HP(X) - HP(X\Y) - H ^ X ) - HP+1(X) - • • • (6 ) where Ну (X) is the (p+l)-th group of cohomology of X with support in Y [13] . When X\Y and Y are topological manifolds of, respectively, dimensions n and m, the following diagram is commutative: HP(X\Y) — ----- HP+1(X) the homomorphism p is a generalization of r and will be called the cohomolo- gical residue homomorphism. IAEA -SM R -11/17 83 1. 3. Relations between the exact sequences (5) and (6) [10] Let X be a reduced complex analytic space of complex dimension n and Y be any closed subspace of X, then we have the following commutative diagram: p P П p+1 (C) • • • - H (X, (С) - H(X\Y,(C) ^ Hy (X, m(x) in(u) j. n(Y) (7) 2. INTERPRETATION OF THE MORPHISMS OF (7) WHEN codim,,. Y= 1 X denotes a complex analytic (reduced) space and Y a closed subspace of X of pure codimension 1. Note that holomorphic functions, C” (or smooth) differential forms and currents can be defined on X (see Refs [3,10] ). 2.1. Case n= 1; principal values; residue current First consider the case where X is a co-ordinate domain U of a Riemannian surface and let ш be a meromorphic form of degree 1 on U having only one pole P in U. Choose the co-ordinate z on U such that z(P) = 0. Consider the current и on U\{P} defined in the following way: for every ф ей(и\{Р} ), let и [ф] = j ' ил.ф и Then Vp(u)[^] = lim / ъ>Лф € ~ * 0 J |z|ae for every фЕ<3(U) is a current on U, whose restriction to U\{P} is u; this current is independent of the choice of the co-ordinate z and of the represen tation of the meromorphic form u; it is the so-called "Cauchy principal value" of Ш. M oreover, dVp(u) = d" Vp(u) = 27ri7Ôp + d'B 84 DOLBEAULT where 7 is the Cauchy residue of 10 at P, 6p the Dirac measure at P and В a current whose support is in {P} ; if P is a simple pole of 10, we can take B = 0. The current dVp(u) will be called the residue current of u. We shall generalize the construction of the residue current to the case of semi-meromorphic differential forms on a complex analytic space X. 2.2. Principal values 2.2.1. An analytic subset Y of a complex analytic manifold X, of complex codimension 1, has normal crossings if, for every point xeY, there exists a chart (z1(. . . , z ) on a neighbourhood U of x such that TJOY = {xGU; z1 (x )...zq(x) = 0; qSn} A semi-meromorphic differential form defined on a neighbourhood U of x G X is said to be elementary if it has a polar set of the above form U П Y. 2. 2. 2. On a complex manifold X, a differential operator D on the space of currents is said to be se mi- holomorphic if, for every x€X, there exists a chart (U,Zj, . . . ,zn) at x such that, on U, where the a }s are C°° functions on U [33] . D operates also on the space ÿ(X) of the sem i-m erom orphic form s on X. Moreover, let A be the ring of the semi-holomorphic differential operators, then 5^(X) and S>'(X) are Д-modules [33]. 2. 2. 3. Let (U, z1, . . . ,z n) be a chart of a complex analytic manifold X and Y be an analytic subset of U whose equation is zx. . . zq = 0 (qâ n). Let be the space of semi-meromorphic forms on U of which Y is a polar set and which are written ar/zP where a is C” on U and p= (p . . . , pn)GNn with Pq+k = O f o r k S l. 2. 2.4. In the above notations, there exists a unique operator T: '(V) such that: (1) If wej/Y has local integrable coefficients, then T(u)= 10, the current defined by и, (2) T is Д - linear. Moreover, T is equal to the Cauchy principal value (see sub-section 2.1.) with respect to each of the co-ordinate functions [6,7,8]. 2. 2. 5. With the notations of 2. 2. 3. , let g = zbgQ be any function where b j . . . b4 =/= 0 and where g0 is holomorphic and without zeros on U. Then, for every cpe^> (U), the formula IAEA -SM R -11/17 85 Vp(u)[m] = lim UA ф à~*0 I (8) defines the current T of 2. 2. 4. This definition of Vp(u) is due to Herrera-Lieberman [10] . One establishes the existence of Vp(u) for a particular representation of to and one verifies that Vp(u) satisfies the conditions (1) and (2) of 2. 2.4. So, we obtain the independence of Vp(u) with respect to the repre sentation of ш and with respect to g; moreover, Vp(u) satisfies conditions (1 ) and (2) of 2. 2. 4. (The idea of this proof is due to Robin [7, 8] ). 2. 2. 6. [10, 6,7]). Let X be a reduced complex analytic space of complex dimension n and let Y be a complex analytic subset of X of codi mension 1, containing the singular locus of X, globally defined by an equation g= 0 where g is holomorphic over X; let f be a holomorphic function over X such that f = 0 implies g= 0; one considers the semi-analytic set (X (>6 )) = {x eX ; |g(x)| >6} Let I[X(>6)] be the integration current over X defined by the semi- analytic chain [X(>ó), e(>ó)] where e(>6) is the fundamental class of (X (> 6)). For a semi-meromorphic form и = a/f (where a is C“ over X), we set Vp(u)[cp] = lim I [X (> ó )] (илф) (9) 6 -о for every ipG®(X). Let ir:X'~>X be a m orphism of Hironaka such that X' be a manifold and Y 1 = 7Г ■! Y an analytic subset with normal crossings (then ir is proper and 7ГI X1 \Y 1 is an analytic isomorphism: X'\Y'->-X\Y) [15]. Such a morphism exists locally. We have: Ур(ш)[ф] = lim l[X '(> ó )] [7г*(иЛср)] = Vp (7г*и) [ тг'ср] (10) Ó-* О where (X'(>6)) = {x'E X 1; | 7r*g(x')|>6} . For ir fixed, Eq. (10) is independent of the representation of и and of g from 2. 2. 5. ; so, Eq. (9) makes sense and, from its expression, is independent of ir. Using a partition of unity, Vp(u) is defined for every semi-meromorphic form и over a reduced complex analytic space X. Let^(X) be the vector space of semi-meromorphic differential forms over X, then Vp:y(X) -*• <£>'(X) is (D-linear. 2. 2. 7. [8]. The ring Д of semi-holomorphic differential operators can be defined on X and it can be shown, at least when X is a manifold, that Vp is a Д-linear application of Д-modules. 86 DOLBEAULT 2.3. Residue current [10, 6, 7] We consider the following residue operator Res = d Vp - Vp d (ID In the notations of 2. 2. 6. , it has the following local expression Res (ш) [ф\ = lim l[W (= ó )] (шЛ. for every фЕ0(Х) and where I[W(=6)] is the integration current on the semi- analytic set lx£X ; |g(x)| = 5} with convenient orientation. 2.4. Interpretation of the diagram (7) [10] 2.4. 1. Denote by the sheaf complex of C°° differential forms on X, by<á^' (# Y) the sheaf complex of semi-meromorphic differential forms with polar set in Y andbysQ^ the quotient sheaf such that the following sequence is exact: °^x 1 sQX - 0 where i denotes the inclusion. 2. 4. 2. Let be the sheaf complex of currents on X an d^y »» the subsheaf of currents with supports in Y; the following short exact sequence defines g>' •X/Y” o-®: r-~ K - K , r - - 0 Principal values and Res define sheaf homomorphisms Vvp- n - ¿ fp ( * Y M ) -» 2n-p, ' X Res: ^ p(*Y)-^>, 2П-Р-1, Y*° In fact, Res (u) depends only on the class of и modulo , hence a homomorphism Define Vp' = j • Vp where j -*■ &' x /y °° ; from Eq. (11), we get dRes = -d Vpd = -Res d, hence a skew complex homomorphism Res: sQ x -*®.'y « IAEA-SMR-11/И 87 Then we get the following diagram of cohomology • • • - HP(X, Sx)^ HP(X, <^(*Y))-* HP(X, sQ'x ) -» ••• (A) (*) V*Vi V p'i Res (*) 4 » ) ^ (B) where V is defined by integration over X; Vp1 and Res are induced by Vp1 and Res respectively; the squares of this diagram are commutative, except (#) which are anticommutative. 2. 4. 3. Finally, we have the diagram (12) where the arrows (A) -*■ (C) and (B) ->■ (D) come from morphisms of de Rham's theorem in cohomology [3] and homology [10]. The squares are commutative except one over three in (A) -» (B) and in (A) ->■ (C) which are anticommutative. Moreover (B) -* (D) is surjective. This diagram gives an interpretation of the residue homomorphism for homology or cohomology classes in X\Y which can be defined by restriction of semi-meromorphic differential forms. Moreover, Herrera-Liberman [10] have a similar result in which the exact sequence (A) is replaced by the corresponding exact hypercohomology sequence for meromorphic differential form s with polar set on Y. 2.4.4. When X and Y are manifolds, then the morphisms of diagram (12) are all isomorphisms and, if res(u) denotes Leray's residue class of the semi-meromorphic differential form w, then Res (u) = 27ri l[Y]nres (w). When X\Y is a manifold, Vp1 splits canonically. When X is a manifold, V is an isomorphism, Res and Vp' split canonically. 2. 5. Interpretation of 6^ for semi-meromorphic differential forms with polar set in Y is given by m eans of homology of open neighbourhoods of Y, with application to the residue of a Cousin data in ([5a], 5, 6]) 88 DOLBEAULT 3. EXACT SEQUENCES (B) AND (D) WHEN X IS A MANIFOLD [32] ; RESIDUE CURRENT IN ANY DIMENSION [31] 3. 1. Let X be a real analytic manifold and Y a closed analytic subset, then the following exact sequences are isomorphic: Hqr(X,Æ>'x) - Hqr (X, Æ>!x/T- ) - H Г (X, S>'. y«°) (В) xj. X / Y ; Y 1 H (X, where the т 's are the homomorphisms of the homological de Rham's theorem. 3. 1. 1. Sketch of the proof. t x is an isomorphism since X is a manifold. (a) It suffices to prove the theorem locally; (b) the theorem is true when Y is a closed submanifold of X (use a homotopy formula for currents); (c) the theorem is true when Y is a hypersurface with normal crossings (use the exact sequence of Mayer-Vietoris valid for closed sets in regular position (Lojasiewicz [21], Chapter 7, prop. 1.4)); (d) to obtain the general case, use a morphism of Hironaka [15] 7r:X->-X, where тг-1 Y = Y has normal crossings. Consider the following diagram: НаГ(Х,^>'.~ „„) а НаГ(Х ,^ '. ») 4 X / Y --- ► 4 X / Y T ~ i T I X \ Y X \ Y Hq(X\Ÿ, Œ) Hq(X\Y, (C) where a and (3 are induced by n . From (с), t ~ is an isomorphism, hence also t„ », ; )3 is an isomorphism since 7r|x\Ÿ is an isomorphism; to show that tx is an isomorphism (which will establish the theorem) and since F(X, ¿Ь''х/y") is made of the currents on X\Y which have an extension over X, it is enough to prove that 7Г*: Г(Х, ^>'J - r(X ,^' ) ' x -x is surjective. It suffices to prove that 7r*: ,ÿn(X) -► ^ n(X) is injective with closed image; this can be done by using the division of a distribution by a real analytic function (Lojasiewicz [21], Chapter 7). 3. 2. Residue current on a real analytic manifold [31] Let X be a real analytic manifold of dimension m, denumerable at infinity and let Y be a closed semi-analytic subset of X of dimension I 3. 2. 1. Let ^.'x be the sheaf complex of the semi-analytic chains on X. We define 'ÿ'.'y” and ^ '.x/ y” as f°r currents. Integration of smooth differen tial forms on semi-analytic chains defines a morphism of sheaves I : « V ^ ' x A current T of dimension q on X \Y is sa.id to be locally bounded on X if, for every Yël^, (X, ), the distribution T A j''~7 is a bounded measure on X\Y (j denotes the inclusion: X\Y -*X). It is a necessary and sufficient condition for the existence of a 0-continuous extension T of T on X (this notion is due to Lelong [20] ). Let Bq(X, Y) be the space of currents T on X \Y of dimension q such that T and dT be locally bounded on X; we define the morphism residue- current by res T = dT - dT; we have: res dT = -d res T. Now, we can write the following commutative diagram (where res is induced by res): Hm'q Г (X, H q Г ( X , ^ ' - Г 4 Н ^ П Х , ^ - ) 7Г i Yi H q(X, The isomorphism is de Rham's for manifolds and irY ' Iy is an isomorphism. 3. 2. 2. A differential form will be said locally integrable if its co efficients are locally integrable. 3. 2. 3. Every oe Hq(X\Y, CD) contains a (m-q)-form which is smooth and closed on X\Y and having the following properties: (i) there exists a locally integrable form cp on X whose restriction to X \Y is cp; (ii) the residue-current of cp is a sem i-analytic chain on Y. The proof uses diagram chasing and the following lemma: 3.2.4. On the complex £>"{X) of currents on X, there exist continuous linear endomorphisms A and R with degree -1 and 0 respectively such that: (i) for every current T, we have T = dAT + AdT + RT; (ii) RT is smooth; sing supp AT С sing supp T; (iii) if T is О-continuous, then AT is a locally integrable form. 4. LERAY's THEOREM In this paragraph, X is a complex analytic manifold of dimension n and Y is a closed analytic subset of X of pure codimension к г 1. 90 DOLBEAULT 4. 1. Case к = 1 4. 1. 1. [28, 28a] Suppose that Y is an analytic subset with normal crossings, then the theorem of Leray is true, i.e. for any class of cohomology a in HP(X\Y, (E), there exists a p-semi-meromorphic form on X whose polar set is Y with multiplicity one and whose restriction to X\Y belongs toa. The proof uses the following lemmas and a classical result of sheaf theory: (a) Let ux be an elementary form around x£X; (Zy, . . . , zn) be local co-ordinates such that the polar set of ux is contained in Zj.. . zn = 0. Let I = (in, . . . , iJ e il, ,n}; wë set z, = z, . . . z¡ and dz, = dz. A . . . Adz. , then J 1 * 1 Ak 1 1 к u = ф + d x where ¡í is elementary and where V = (dz, /zjJA^j + > iep([l...... n}) with (//j and AC in a neighbourhood of x. (b) Let ux be a germ of closed elementary form; (i) if degree ux > dim^-X, then ux is exact; (ii) if degree wx S dimŒX, then ux is cohomologous to a germ of meromorphic form. 4. 1. 2. [28]. The above result is valid for meromorphic forms if X is a Stein manifold. 4. 1. 3. Recall the following theorem ([14] , theorem 2) whose proof uses Hironaka's resolution of singularities where f:X\Y-*X is the; canonical injection and f2'x(*Y) is the sheaf of mero morphic forms on X with polar set contained in Y. 4.1.4. Using theorem 4.1. 3. , Grothendieck proved a theorem ([14], Corollary) whose particular cases are: If X is a Stein manifold, or if X is a projective algebraic manifold and Y is the support of an ample positive divisor of X, then there exists a canonical isomorphism: НЧГ (X, f i '( * Y ) ) H 4(X\Y, <0 X x 4.1.5. Using theorem 4. 1. 3. and lemma (b) of 4.1.1., Robin proved [28] (compare also 2. 4. 4. ): Let Y be any analytic subset of codimension 1 of X, then any class of cohomology of HP(X\Y) contains the restriction to X\Y of a closed semi- m erom orphic form on X whose polar set is contained in Y (but with no conditions on the multiplicities of the poles). IAEA-SMR-11/17 91 4. 2. General case (к г 1) [29] (see Refs [23, 24, 34] ) 4. 2. 1. Kernels. Let ly be the integration current defined by Y. We call kernel (or quasi-kernel., respectively) associated to Y every (2k-1 )-differential form К locally integrable on X, with singular support in Y such that ly-dK is a smooth (or locally integrable, respectively) form on X. 4.2.2. Existence of kernels. Let X be a smooth real manifold which is denumerable at infinity and Y be a closed subset of X. Then, for every current T such that: (i) T is 0-continuous; (ii) df~^ 0; (iii) supp sing T C Y , there exists a differential form К locally integrable on X, with singular support contained in Y and such that T - dK is a smooth differential form on X. 4. 2. 3. A differential form в defined on an open subset U of X is called quasi-smooth if 6|u\Y is smooth and if в and d6 are locally integrable on U. Let£f"x.Ybe the sheaf complex of quasi-smooth differential forms on X. 4. 2. 4. The proof of 4. 2. 2. is a consequence of the following two lemmas: (a) For every germ of current T of degree p ï 1, 0-continuous and d-closed, there exists a germ of (p-l)-differential form K, locally integrable, smooth outside the singular support of T and such that T = dK (this is a Poincare lemma). (b) The inclusions S'x -*-¿2^x.Y ^x induce isomorphisms in cohomology НрГ(Х,<Гх) 2 НРГ (Х ,^ Х;Т) 2 HPr(X,¿Z>'x) 4.3. General case (continuation) : quasi-simple forms [29] 4. 3. 1. Let £ P(X;Y) be the subspace of T(X \Y , <£х) of the p-form s such that ф = (КАф + 0)| X \Y (13) where К is a quasi-kernel associated to Y, Ф a (p-2k +1 )-smooth form on X and в a quasi-smooth p-form on X; this definition does not depend on the choice of K. Those form s are said quasi-sim ple on X. 4 .3.2. Let res: EP(X;Y) -» Г (Y, s \ 2k+1) be the morphism which, in the notations of (13), is defined by res cp = i"ip (14) where i is the inclusion: Y-X; rescp does not depend on the expression of cp and is called the residue-form of cp. For every cpe£p(X;Y), we have dcp£ EP+1(X;Y) and res (dcp) = -d (rescp). 92 DOLBEAULT 4. 3. 3. ц being the restriction morphism, then the following sequence is exact О-Г(Х,£Гх.у) Ü r/(X;Y)-S r(Y ,^‘Y)- 0 Then we have the following diagram ------► Hpr( X ,^ 'X;Y) £ HP£(X ;Y ) (15) where the squares commute, except (*) which anticommutes; the vertical morphisms are defined by integration over X, X\Y and Y, respectively. 4.3.4. From de Rham's theorem for analytic spaces [3], we have m orphism s e : HP(Y,(E) - HpF(Y,e?‘Y) which have canonical retractions I: HP r(Y, tf’y) -* HP(Y, The following diagram is commutative (16) (Y, where t y comes from de Rham's theorem in homology. From Eqs (15) and (16), we get 4. 3. 5. The following two conditions are equivalent: a e H2n (X\Y, Œ) contains a closed form ф e EP(X;Y) (i) 6.. a = [Y] П|3 where )3 G Hp"Zk+1(Y, 4. 3. 6. A more precise theorem can be obtained, but without a diagram of the form (15). Namely, К being a kernel associated to Y, let ff"^(X;Y) be the subspace of £P(X;Y) made of the smooth p-form s ф on X \Y such that ф = (КЛ^ + б)| X \Y where Ф and в are smooth on X. The form s of (^^(Х;У) are said to be K-simple on X. IAEA-SMR-11/17 93 4. 3. 7. In the above notations, the following two conditions are equivalent: (i) aeH 2n.p(X\y,([) contains a closed p-form <р€5^^(Х;У); (ii) there exists )3GHP 2k+1(Y,(C) such that = [Y]n(3. Condition (ii) is not always satisfied as can be seen for: X = Œ2 and Y = {(zj, z2)e(C2 ; ZjZ2 = 0} . The proof of the theorem essentially uses the following result: Let ¡3 G Hp‘2k+1(Y, Œ) such that = 0 (i being the inclusion: Y-*-X), then every u 6 r(Y ,|,pY'2W) belonging to e)3 is the residue -form of a closed form cpe 4. 3. 8. Corollary: Let Y be a submanifold of X, then every element of H2n_p(X\Y, (E) contains a closed form belonging to CT£(X;Y). If Y is of codimension 1, with local equation s = 0 and if the kernel К is constructed from the local kernels (1/27ri) (ds)/s, then 4. 3. 8. is the classical theorem of Leray. 4. 3. 9. Remark: In theorem s 4. 3. 5. and“4. 3. 7. , the closed p-form cp is considered to be a current of dimension 2n-p and, in this way, defines a class of homology. 5. COMPOSED RESIDUES They have been defined and studied previously in ([18, 23, 35] ). 5. 1. Mayer-Vietoris morphism [30] 5. 1. 1. Let X be a paracompact real analytic space, Yj, Y2 be closed semi-analytic subsets of X; let Y = Yj U Y2 and y = Yj П Y2. The Mayer-Vietoris morphism /j* is defined in the exact sequence Hq(y,Œ) - Hq(Yj, (E) © Hq(Y2,Œ) - Hq(Y, Œ)"-* and it is related to the residue homomorphism 6% and the composed residue homomorphism 6.J.2 defined as in Leray's papers [18,35] by the following commutative diagram: Hq(X\Y,(C) 6* ; Hq-].(Y, Œ) — 5.1. 2. Interpretation of by means of semi-analytic chains. Lemma. Let be the sheaf complex of semi-analytic chains on X, then the following sequence is exact: 0 - Г (XM'x) “ Г ( X x) © Гу2(Х,^.'х) ® rY(X,^.'x)- 0 (17) 94 DOLBEAULT where a= i{ - i^; P = ii+i2; i'k and ik being defined by the inclusions: y^ Yk and Yk->Y (k = 1,2) respectively; (this comes from r(Z,çg.’z) = r z(X, eg'.x) if Z is a closed semi-analytic set of X). 5.1.3. From Ref. [3] , theorem 2. 8, we obtain the result: The following diagram where vertical morphisms are isomorphisms is cummutative : • • — Н Г(Х ,«'х)- H rYi(X,^'.x) © HqrY2(X,^'.x)- HqrY(X ^ '.x) i * i (18) - Hq (y, Œ) — Hq(Y1;Œ) © Hq(Y2,Œ) - Hq(Y,Œ)------ 5. 1.4. If X is a manifold, the semi-analytic subsets Yj and Y2 being in regular position, we have the analogous of (17) for currents ([21 ], chapter 4, proposition 1.4.) and the isomorphism of exact sequences of 5. 1. 3. can be factorized through a morphism from the exact sequence (18) to be corres ponding exact sequence for currents. 5. 1. 5. Remark: This theory can be done for any finite number of closed semi-analytic subsets of X. 5.2. Multiple residues [11] 5. 2. 1. Let X be a complex analytic space of dimension n and Y0 , . . . , Y$ (s+1) complex analytic subsets of X of codimension 1; let Y =n Y¡ (i e i = (0 ,,.,,s) ). Let Д1 be the set of the t-simplices S ~ {iQ, . . . it} С Д and $^(*YS) the sheaf of the q-semi-meromorphic differen tial form s on X whose polar set is contained in Ys = U (Y¡; i G S). We define the bigraded sheaves Ct,q(0 § t S s; 0 5 q 5 n) as the sheaves of t-Cech cochains with values in the ^ X(*YS) in the following way: for every open set W in X Ct,q(W)= H tr ( w ,^ Ys )) S e Д C*’* is a double complex, we consider the total complex e-£e- Ш with t + q = m The sheaf S is a subsheaf of С IAEA-SMR-11/17 95 We define the sheaf complex Q’ by the short exact sequence 0 - S' - c’ - Q' - 0 (19) The exact cohomology sequence (A1) associated to (19) is analogous to the sequence (A) (2.4. 2. ), and using integration, we can prove that there exists a morphism (A1) -* (C) which is an isomorphism if X is a manifold (in fact, one square over three is anticommutative). 5. 2. 2. Suppose now that Y is a complete intersection; Д= (0,. . . , p-1) and codim Y = p, then we can define iteration of Vp and of Res of 2. 2 and 2. 3. and prove the existence of the diagram analogous to the diagram (12) (2. 4. 3. ). The multiple residues defined in that way are related to the composed residues and in the case of a complete intersection Y give interpretations of the residue homomorphisms. The present paragraph contains a work by Colef to be published. As in Ref. [10] , one can use uniquely holomorphic and meromorphic differential forms instead of C“ and semi-meromorphic forms and then replace cohomology by hypercohomology. The multiple residues are useful to construct the dualizing complex used in the proof of the duality theorem for analytic spaces (see a forth coming note by Herrera- Ramis-Ruget and the paper of Ruget, Complexe dualisant et résidus (Journées de géométrie analytique, Poitiers juin 1972). For the theorem of duality, see the paper by Ramis, Ruget and Verdier, Dualité relative en géométrie analytique complexe, Inventiones math. 13 (1971) 261. 6. PROBLEMS 6. 1. To generalize Poly's results using kernels (4. 2. , 4. 3. ) to analytic spaces. 6. 2. What about the growth of kernels in the neighbourhood of Y? 6. 3. To find a more complete real analytic theory of residues. 6. 4. To study meromorphic and semi-meromorphic differential forms with poles of multiplicity one and their generalizations in the sense of Poly (4. 3. ); in particular, to what extent does the theorem of Leray remain valid for Y (and X) with singularities? (compare with Robin's results (4.1)). 6. 5. To construct a semi-meromorphic differential form with given residue current (partial results have been obtained; see Ref. [6] n. 4. 3, and Ref. [7] , exposé du 28. 1. 70, n. 5. 2. ) 6. 6. When X is a complex analytic manifold and when Y is a particular case of equisingular analytic subset of codimension 1, Vp(u) and the residue current of и (for a closed meromorphic form u>) have been defined in Ref. [5] , chapter 4,D); when Y has normal crossings, results have been given in Ref. [7] , 28, 1. 70, n. 4. 1. This study of the residue current has to be made in the general case. 96 DOLBEAULT 6.7. To find new applications of the existence theorem of kernels (4.2.2). Among the problems given in Ref. [7], 28.1.70, n.5.3., nn.2,3,4,5 are now solved. REFERE NCES [1] ASADA, A., Currents and residue exact sequences, J.Fac.Sci.Shinshu Univ. 3 (1968) 85. [2] ATIYAH, M.F., HODGE, W.V.D., Integrals of the second kind on an algebraic variety, Ann. Math. 62 (1955) 56. — [3] BLOOM, T ., HERRERA, M., De Rham cohomology of an analytic space, Inventiones Math. 7 (1969) 275. [4] BREDON, G.E., Sheaf Theory, Mc-Graw-Hill Series in Higher Mathematics (1967). 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[21] MALGRANGE, B., Ideals of differentiable functions, Tata Institute, Bombay, No.3, Oxford Univ. Press (1966). [22] NORGUET, F ., Sur la théorie des résidus, C.R. A c.Sci.Paris 248 (1959) 2057. [23] NORGUET, F., Dérivées partielles et résidus de formes différentielles sur une variété analytique complexe, Sém.P. Lelong, 1958-59, No.10 (24 pages). [24] NORGUET, F ., Sur la cohomologie des variétés analytiques complexes et sur le calcul de résidus, C.R. Ac. Sci. Paris 258 (1964) 403. [24a] NORGUET, F ., Sém .P.Lelong (1970) Lecture Notes 205. [25] NORGUET, F ., Résidus et q-convexité. Colloque sur les fonctions de plusieurs variables complexes, Paris (juin 1972). [26] KING, J.R., A residue formula for complex subvarieties, preprint 1972. [27] POINCARE, H ., Sur les résidus des intégrales doubles, Acta Math. 9 (1887) 321. [28] ROBIN, G., Formes semi-méromorphes et cohomologie du complémentaire d*une hypersurface d’une variété analytique complexe, C.R. Ac.Sci.Paris 272 (1971) A-33-35. [28a] ROBIN, G., Sém.P.Lelong (1970) Lecture Notes 205. [29] POLY, J.B., Sur un théorème de J. Leray en théorie des résidus, -C.R.Ac.Sci.Paris 274 (1972) A-171. [30] POLY, J.B., Morphismes de Mayer-Vietoris et résidu composé, à paraître. [31] POLY, J.B ., in Thèse en cours. [32] POLY, J.B., Cohomologie locale et courants. Journées de Géométrie analytique, Poitiers (juin 1972). [33] SCHWARTZ, L., Courant associé à une forme différentielle méromorphe surune variété analytique complexe, Coll.int.C.N.R.S., Géométrie différentielle, Strasbourg (1953) 185. [34] SHIH WEI SHU, Une remarque sur la formule de résidus, B ull.A m er.M ath.Soc. 76; (1970) 717. [35] SORANI, G., Sui residui delle forme differenziali di una varietà analitica complessa, Rend. Mat. 21 (1962) 1. IAEA-SMR-11/18 ON CONNECTIONS S. DOLBEAULT Department of Mathematics, University of Poitiers, Poitiers, France Abstract ON CONNECTIONS. This paper covers the following topics: — Preliminaries: vector fields on a manifold, Lie groups; connections in a principal fibre bundle: subspaces of tangent spaces, connections, parallelism, curvature and torsion, linear and affine connections, metric connections; connections and characteristic classes: Weil homomorphism, Chern classes. The purpose of this paper is to give an elementary survey of the classical theory of connections. This theory arises from papers of Cartan [2, 3]; it was then developed by Ehresmann, [8, 9], and it has been the subject of many books among which we shall only refer to those of Chern [ 7] , Lichnerowicz [ 13] and Pham Mau Quan [ 16] . The objects under consideration are differentiable manifolds on which we first study vector fields and the action of a Lie group (see Lichnerowicz [ 14]); we then construct bundles over M, especially principal bundles and vector bundles (for the theory of fibre bundles, see Husemoller [11]). Then, we shall give the geometric aspect of a connection and, on having defined covariant differentiation, we shall introduce curvature and torsion. We shall not speak about holonomy groups. As a conclusion, we shall try to expose the relationship between differential geometry and cohomology on a manifold. To do this, we need characteristic classes. These classes were introduced by Pontryagin, Stiefel and Whitney. They are the object of many papers of Chern [ 6] . Hirzebruch [ 10] gave the axiomatic definition of Chern classes; we shall also refer to a paper by Milnor [ 15] . The relation between connections and characteristic classes results from the Weil homomorphism which was first studied by Cartan [4, 5], and may be used to study the holomorphic section of a Hermitian vector bundle (Bott and Chern [ 1]). F o r the whole of this paper, we refer to Kobayashi and Nomizu [ 12] . The manifolds, mappings, etc. are all supposed differentiable of class C°°, unless stated otherwise. 1. PRELIMINARIES 1.1. Vector fields on a manifold a) Definitions: A diffeomorphism of a differentiable manifold M onto itself is called a differentiable transformation of M, or, simply, a transform ation of M. 97 98 DOLBEAULT A 1-parameter group of transformations of M is a mapping of IR X M into M: R X M -----►M (t,x) I-----► b) Properties: conversely: Proposition 1: Let X be a vector field on a manifold M; for every point x0 of M, there exists a neighbourhood U of xQ, a real number e>0, and a local 1-parameter group of local transformations tpt : U -» M, |t| < e, which induces the given X. The proof is given by solving a differential equation x(t) = X , where x(t) = %(хо) and the initial condition x0 = (?. X )x = ( ( The automorphisms tp, and tp" are related by v>*(( Proposition 2: Let tp be a transformation of M; if a vector field X generates a local 1-parameter group of local transformations tpt, then the vector field tptX generates tpo tpt о tp'1. Corollary 2: A vector field X is invariant by tp (i. e. tptX = X) if and only if tp commutes with tpt. Remark: These considerations permit a geometrical interpretation of the bracket of two vector fields, which can also be used as a definition [ 16] : Proposition 3: Let X and Y be two vector fields on M. If X generates a local 1-parameter group of local transformations tpt, then the bracket Z = [X,Y] is the vector field: IAEA-SMR-11/18 99 Z — lim Y (Y - (^).Y) t - о (i >) more precisely, at xeM, Z, lim X x - ((Ы у)х)] = 0 t 0 >) 1.2. Lie groups a) Definitions: A Lie group is a group G which is also a differentiable manifold and which satisfies the following condition: the group operation GX G -----►G (a, b) I-----► ab"1 is a differentiable mapping of G X G into G. We denote by La (resp. Ra) the left (resp. right) translation of G by an element a of G, i.e . La x = ax (resp. Ra x = xa), VxeG For aeG, ad a is the inner automorphism of G defined by (ad a)x = ax a"1, VxeG; the set of all such automorphisms is a group denoted by ad(G) and called the adjoint group of G. b) Action of a Lie group on a vector field. We suppose that X is a vector field on G; we say that X is left invariant (resp. right invariant) if it is invariant by all left translations La (resp. right translations Ra), aeG. We define the Lie algebra g of G to be the set of all left invariant vector fields on G, with the usual addition, scalar multiplication and bracket operation; as a vector space, g is isomorphic with the tangent space Te(G) at the identity, the isomorphism being given by the mapping which sends Xeg into Xe (Xe is the value of X at e); thus, n being the dimension of G, g is a subalgebra of dimension n of the Lie algebra of vector fields8 C (G ). Every Aeg generates a (global) 1-parameter group of transformations of G [ 14] . Indeed, if a t = then a.t+s = a, t a s V (s ,t)e E X IR; ' we call a( the 1-parameter subgroup of G generated by A. 1 0 0 DOLBEAULT Another characterization of at is that there is a unique curve in G such that its tangent vector ât at at is equal to LatAe, and a0 = e; in other words, a is the unique solution of the differential equation with initial condition Denote a 1 = <рг(е) by exp A; it follows that exp tA = a( VteIR The mapping A -» exp A of g into G is called the exponential mapping and we have: Let срЪе an automorphism of the Lie group G; it induces an automorphism cp, of the Lie algebra g of G with the following properties: if A is an element of g, then In particular, for every a£G, the automorphism ad a which maps x£G into axa"1 induces an automorphism of g, also denoted ad a. The represen tation a -» ad a of G is called the adjoint representation of G in g. For every aeG and A eg, we have (ad a) A = (Ra-1)eA, because A is left invariant. Let A and В be elements of g and let [B, A] = lim 0- (( This follows from proposition 3 and from the fact that g is a set of left invariant vector fields. c) Action of a Lie group on differential forms. A differential form a is said left invariant if (La)'# = a, VaeG. The set of all left invariant 1-form on G is a vector space g” which is the dual space of g (i.e. for Aeg and neg’, the function o(A) is constant on G). Let do be the exterior differential of the differential form a; if a is left-invariant, so is also da, and, in particular, we have aeg’ do (А, В) + I о ([ A, B] ) = 0 j A, Beg IAEA-SMR-11/18 1 0 1 This equality is known as the Maurer-Cartan equation [12, 16] . Definition: The canonical 1-form в on G is the left-invariant, g-valued 1-form satisfying to 0 (A) = A VAeg. This condition determines в uniquely. It is often useful to have an expression of в in local co-ordinates. Let ej, . . ., en be a basis for g; we can set n в = ^ e. i= 1 we introduce the structure constants Cl, by setting " jk i = 1 then the Maurer-Cartan equation takes the form: n d e1 + i c jk 6j Л 0k =0 (i = 1, . . ., n) j.k = l Remark: the set { : is a basis for g°. d) Lie transformations groups [ 14] : Definition: We say that a Lie group G is a Lie transformations group on a manifold M, or that G acts (differentiably) on M if the following conditions are satisfied: (1) every aeG induces a transformation of M x ---- »xa VxeM (2) the mapping GX M -----» M (a, x) I ► xa is a differentiable mapping. r VxeM (3) x(ab) = (xa) b, J [ V(a, b)£G X G We also write Ra x for xa and say that G acts on M on the right. We say that G acts effectively (resp. freely) on M if Ra x = x for all xeM (resp. some xeM) implies that a = e. Suppose that G acts on M on the right, consider an element A of g, and associate to A, for t real, the set a( = exp t A 102 DOLBEAULT This set is a 1-parameter subgroup of G and its action on M induces a vector field A*. We have the proposition: Proposition 4: Let a Lie group G act on the manifold M on the right. The mapping: a : g ----► SC ( M) Al----►A' is a Lie algebra homomorphism. If G acts effectively on M, then a is an isomorphism of g into S f { M). If G acts freely on M, then, for each non-zero Aeg, a (A) never vanishes on M. 2. CONNECTIONS IN A PRINCIPAL FIBRE BUNDLE 2.1. Subspaces of tangent spaces Let P = P(M, G) be a principal fibre bundle on a manifold M, with structural group G and projection p; for ueP, let Fu be the fibre through u. Consider the tangent bundle T(P) of P and the set TU(FU) of vectors tangent at u to the fibre Fu. Suppose that we can choose in TU(P) a set TU(P) of vectors satisfying the following conditions: (a) TU(P) is the direct sum of TU(FU) and TU(P). TU(P) = TU(FU) + T„(P) (b) The correspondence u -» TU(P) is invariant by G. As G acts on P by right translation Ra u = ua, aeG, this can be written: Tua(P) = (Ra)„Tu(P), VueG, VaeG. (c) TU(P) depends differentiably on u. When such a decomposition is possible, we shall say: Definition: Tu(-Fu) is the vertical subspace of TU(P) at u; TU(P) is a horizontal subspace of TU(P) at u. Given TU(FU) and TU(P), every vector XueTu(P) admits a unique decomposition: Xu = 'Xu + b l where IXue T u(Fu) and X ue T u( P ) . |XU and Xu are, respectively, the vertical and horizontal components of Xu. To have results about existence of fields of horizontal vector subspaces, let us consider a subset N of M; suppose we can define a field of horizontal vector spaces on N by associating to every ueP, such that p(u)eN, a horizontal vector subspace TU(P) of TU(P). We have the following: Theorem 5: If M is paracompact, every field of horizontal vector spaces on the restriction of P to a closed subset of M can be extended in a field of horizontal vector spaces on P. If M is compact, P admits a field of horizontal vector spaces. Proof is based on the notion of partition of unity. IAEA-SMR-11/18 103 2.2. Connections a) Definitions: If we can define a field ofhorizontal subspaces in T(P) we say that we have a connection Tin P. Given a connection Tin P, we associate with it a 1-form и on P, with values in g as follows: For each vector field XeT(P), we define w(X) to be the unique Aeg such that A’ = IX. The form u is called the connection form of Г. b) Properties of the connection form: 1) (a) VAeg, u(A* ) = A; in particular u(X) = 0 < = > X is horizontal. Ф) For every aeG and every vector field X on P, u((Ra)eX) = adía"1) и (X). Proof: Property (a) follows directly from the definition of u. To prove (13), one may consider every vector field X as the sum of IX and X and study separately each of these two components. 2) Conversely, let и be a 1-form on P, satisfying (a) and (0); then u define a unique connection Гоп P for which the connection form is u. Proof: This property follows directly from the relation between forms and vector fields. 3) Translating theorem 5 in terms of connections, we have the following proposition about existence of connections: Proposition 6: И N is a closed subset of a paracompact manifold M, every connection on the restriction of P(M) to N can be extended in a connection on P(M). If M is compact, P(M) admits a connection. 4) If P(M) admits a connection, we shall see later (2.4, a, example) that it admits other connections. 5) Local co-ordinates: Let {Ua} be an open covering of M, with a family of isomorphisms : P '1(Ua ) ---- ~ U a X G and the corresponding family of transition functions: ф , : U П U„ ---- » G For each a, let sa : Ua -* P be the cross-section on Ua , defined by -x \ XGU«’ sa(x) =|//â (x, e) e is the identity of G 104 DOLBEAULT Let в be the canonical 1-form on G. In a natural way, we associate to в a g-valued 1-form on Ua П Ug / <¡¡ by setting ваВ = < 6 9 for each a, define a g-valued 1-form ua on Ua by setting ioa = s„ u then we have: Proposition 7: On Ua П Ug, the forms Qa g and are subject to the conditions: ug = ad (фаg ) ioa + вав Conversely, given the canonical form в and a family of g-valued 1-forms {toa}, each definedon UH and satisfying the precedent condition on Ua П Uj, then, there is a uniqueconnection form ш on P which gives rise to{u)K} in the way described above. 2.3. Parallelism a) Lift of a vector field on M: The projection p : P ----► M induces a linear mapping P • (?) T p(u) (M) when a connection Г is given on P, the projection p is an isomorphism between Тц(Р) and Tp(u) (M). Conversely: Definition: Given a vector field X on M, there exists a unique vector field 2 on P, which is horizontal and which projects onto X. This vector field X is called the horizontal lift (or simply the lift) of X. Remark: The existence and uniqueness follow from the fact that p is a linear isomorphism of TU(P) onto Tp(U) (M). Properties: - The set of lifts of all vector fields on M is invariant by the right translations of G. - For two vector fields X and Y on M: [X, Y] = [X, Y] b) Lift of a curve on M By curve с = {x(t), 0 S t s 1} in M, we shall mean a map IAEA-SMR-11/18 105 which is piecewise differentiable of class C1. A horizontal lift (or simply a lift) of С is a curve g = {u(t), 0 s t a 1} in P which is: с is piecewise differentiable of class C1, с projects on c: p(u(t)) = x(t) V te [ 0 , 1], every vector tangent to с is horizontal. Existence and unicity: given a curve с in M, and a point ujEP in the fibre over x(0); p(u0) = x(0), there exists a unique lift ç of c, starting from u0. (This is nothing but the existence and unicity of the solution of a linear differential equation with given initial condition). Definition: Consider a curve С and its lift ç starting from u0 G P; such a lift ends at иг such that p(uj) = xj. We thus obtain a correspondence between uQ and Up we shall write it c, i.e. C(U0) = Uj Let u0 describe the fibre p"1(x0); we thus obtain a mapping с of the fibre p_1(x0) onto the fibre p '^x j. This mapping с is, in fact, an isomorphism and is called the parallel displacement along the curve c. Remark: Every right translation of G maps a horizontal curve onto a horizontal curve, so that Ra commutes with c: Ra о с = с o Ra; VaeG. It follows that с is an isomorphism between two fibres of P. 2.4. Curvature and torsion a) Tensorial forms. Let p be a representation of the structure group G on a finite dimensional vector space V (i. e. p(a) is a linear transformation of V for each element a of G and p(ab) = p(a) p(b) for each set (a,b) of elements of G.) Definition: A tensorial form of degree r, of type (p,V) on P is a V-valued r-form so n P such that: (i) Ra* a = p(a-1) • a VaeG (ii) a (Xj,. . ., Xr ) = 0 whenever one of the vectors X j,. .., Xr is tangent to a fibre. In particular, if p is the adjoint representation of G in g, a form a satisfying (i) and (ii) is a tensorial form of type ad G. Example: Let и be a connection form on P and a be a tensorial 1-formof type ad G on P; then to = to + a defines a field of subspaces of T(P) which can be taken as a horizontal vector field, so that u defines another connection on P [ 13]. Conversely, if to and u are two connection forms on P, their difference is a tensorial 1-form of type ad G. 106 DOLBEAULT Remark: Let Г be a connection on P and in a point uEP, let h be the projection h : TU(P ) ----»- TU(P) Let a be a V-valued r-form on P satisfying only (i); we associate to a a tensorial r-form of type (p, V), by setting («h) (Xj...... X r) = e (Xi, .... Xr) b) Exterior covariant differentiation. The remark above implies: Definition: given a connection Fin P, we can associate to the operator d of exterior differentiation an operator V defined, for every form a on P, by Va = (da) h which is called operator of covariant exterior differentiation. The form Va is called exterior covariant derivative of a. This definition implies: Proposition 7: Given a V-valued r-form on P, satisfying (i), its exterior covariant derivative is a tensorial (r+ l)-form of type (p, V). In particular, if и is the connection form, Vu is a tensorial 2-form of type ad G. c) Curvature and torsion of a connection. Definition: Given P with the canonical form 9 and a connection form u, we set: Г2 = Vu, © = V0 Í2 is the curvature form of the connection; © is the torsion form of the connection. Properties of the curvature form: Structure equation of E. Cartan: du = - j [u, со] + fî Proof: At ueP, and for Xu, Yue'IJ1(P), write the structure equation: du (Хц, Yu) = - I [u(^), «(%,)] + Y„) both sides of this equality are bilinear and skew-symmetric in Xu and Yu. Since every vector of T(P) is the sum of a vertical vector and a horizontal vector, it is sufficient to verify the last equation in the three following cases 1) Xu and Yu are horizontal: then u (Хц) and u (Yu) are null and the structure equation reduces to the definition of 2) Xu and Yu are vertical: then f2(Xu, Yu) = 0 since Г2 is a tensorial form and the structure equation, analogous to the Maurer-Cartan equation results of the definition of exterior derivation. IAEA-SMR-11/18 107 3) Xu is horizontal and Yu is vertical: for the same reason S7(XU, Yu) = 0; as in the precedent case, it remains, by coming back to vector fields X and Y: 2 du (X, Y) + u [X, Y] = 0 this equation is similar to that of Maurer-Cartan and can be proved in the sam e way. Local co-ordinates. With the notations of section 1, setting n n u = ^ w1 e¡, Í2 = ^ Q1 e¡ i=i i — l the structure equation can be expressed as follows: n du1 = - I ^ cjkuJAuk + ft1 (i= 1,. .., n) j,k=l Bianchi1 s identity: V Í2 = 0. Proof: By the definition of V, it suffices to prove that df2(X, Y, Z) = 0 whenever X, Y and Z are all horizontal, and this follows immediately from the exterior differentiation of the structure equation. 2.5. Linear and affine connections These are two particular, but very important cases. a) Linear connection. Let G = GI (n; IR) and P be the bundle L(M) of linear frames over the manifold M of dimension n. Then the canonical form в is a tensorial 1-form of type (GI (n; IR), ]Rn). A connection in P is then called a linear connection on M. We have then the structure equations of E. Cartan: d0 = - ил0 + 0, du = -u/\u + f2 and Bianchi1 s identities: V© = Í2 а б, V Q = 0 b) Affine connections. Let An be the real affine space of dimension n. The space tangent to M at x can be considered as an affine space AX(M); an affine fram e of M at x consists of a point f of AX(M) and a linear fram e at x We consider the set A(M) of all affine frames of M and define a projection p : A(M) ----- M by setting p(u) = x 108 DOLBEAULT for every affine fram e u at x. It is well known that A(M) can be considered as a principal bundle with structural group A(n; IR) (the group of affine transformations of An). An affine connection Г of M is a connection in A(M). c) Relation between linear and affine connections on M. L(M) can be considered as a subbundle of A(M); namely, Proposition 8: There exists a homomorphism ip: A(M) ----► L(M) This homomorphism maps every affine connection Tof M into a linear connection Г of M. The correspondence Г-» Tbetween the set of affine connections and the set of linear connections of the same manifold M is 1-1. This proposition justifies the fact that one of the words "linear connection" and "affine connection" is often used in place of the other. Remark: Complex linear and affine connections can be defined in the same way by replacing IR by (C. 2.6. Metric connections a) Definitions: Let E be the vector bundle, associated with the principal bundle P, with standard fibre Fn (F = IR or (C) and projection pE . A fibre metric in E consists in giving in each fibre an inner product g such that, if xeM and if ax(x), cr2(x) are cross sections of P, then g(a1(x), cr2(x)) depends differentiably on x. A connection in P is a metric connection if the parallel displacement of fibres of E preserves the fibre metric g, that is: for every curve с ={xt, 0 st sl} in M, the parallel displacement p’^X q) -* p'^Xj) is isometric (i.e. preserves g). b) Proposition 9: If M is paracompact, every vector bundle over M admits a fibre metric g; to such a metric g corresponds a connection Tin P. Proof: The first part results of a partition of unity; the second is a consequence of the first and of proposition 6. c) Examples: 1) We recall that a Riemannian metric on M is a covariant tensor field of degree 2 which is positive definite: g(X, X) g 0 V X e^(M ) g(X, X) = 0 < = > X = 0 IAEA-SMR-11/18 109 and symmetric: g(X, Y) = g (Y, X) jvxe^-(M) [v Y e á '(M ) so that g defines an inner product in the tangent space TX(M). The set (M, g) is called a Riemannian manifold. Proposition 10: Every Riemannian manifold admits a unique metric connec tion with vanishing torsion, which is called the Riemannian connection of M. 2) Let M be a real manifold of even dimension n = 2 m. Generalizing the notion of complex manifold, we define on M an almost complex structure by giving a tensor field J which is, at every point xeM,an endomorphism of the tangent space TX(M) such that J2 = - Identity. The set (M, J) is an almost complex manifold. A Hermitian metric on an almost complex manifold M is a Riemannian metric g invariant by the almost complex structure J: g(JX, JY) = g(X, Y) for any vector fields X and Y. On an almost complex manifold, a complex tangent vector Zx is said of type (1, 0) (resp. (0, 1 )) if there is a real tangent vector Xx such that Zx = Xx - iJXx, (resp. Zx = Xx + iJXx) A vector field of type (1, 0) such that Zf is holomorphic for every locally defined holomorphic function f is a holomorphic vector field. Given an almost complex manifold M with Hermitian metric g, the fundamental 2-form Ф is defined by Ф(Х, Y) = g(X, JY) for all vector fields X and Y, so that $ is also invariant by J. A linear (or affine) connection on M is said almost complex if it is a connection in the bundle C(M) of complex linear (or affine) frames on M. We have the following proposition: Proposition 11: Let M be an almost complex manifold with metric g; if the Riemannian connection defined by g is almost complex, then the fundamental 2-form Ф is closed. 3. CONNECTIONS AND CHARACTERISTIC CLASSES 3.1. Weil homomorphism Let G be a Lie group with Lie algebra g. We consider multilinear sym m etric mappings of order k: fk : g X g X . . . X g ----► В 110 DOLBEAULT invariant by G, i. e. such that VaeG and V ^, t2, ,. ., tk) eg X g X ... X g, fR ((ad a)tb (ad a)t2,. . ., (ad a)tk) = ^(tj, t2,. . ., tk); letSk(G) be the set of such fk and S(G) = ^ Sk(G) k>0 S(G) is a commutative algebra when we define a multiplication in the following way: we first consider fk e Sk(G) and fcG SC(G); we define fk+íe Sk+e (G) by (ti» • • •. t k+c) = "(kTjyr X ‘ ' to (fk + fc ) (f1 + fj ) = fkf‘ + Л-1 + f{ f* + f{ fJ Let P be a principal fibre bundle over a manifold M, with structural group G. Given a connection Tin P, let Г2 be the curvature form and define, for fk G Sk(G), and X.G Тц(Р), (1 s i s 2k) fk (П) (Xj, . . . , X2k) = —Y eof 2k(í2(Xo(1), Xo(2)), .. . , Xo{2k))) О where the summation is taken over all permutations a of (1, . . ., 2k), and e0 denotes the sign of the permutation ст. With these notations, we have the following: Theorem 12: Let P be a principal fibre bundle over a manifold M, with group G and projection p; let ГЬе a connection in P and Г2 its curvature form; then: (i) for each fkeS k(G), the 2k-form fk(f2) on P projects to a unique closed 2k-form, say 7 k(ii), on M, i.e . fk(í2) = p*(Tk (f2)) ii) let w(fk) be the element of the de Rham cohomology group H 9 k (M; Ш.) defined by the closed 2k-form Yk(i3); then w(fk) is independent of the choice of the connection Г and w : S(G) ----► M*(M; K) is an algebra homomorphism which is called the Weil homomorphism. Definition: Let V be a vector space on IR (or (C) and f1,..., fr a basis for the dual space of V. A mapping ф: V -* К (resp. (С) is called a polynomial function if it can be expressed as a polynomial of f1, . .., fr. (This concept is evidently independent of the choice of the basis). IAEA-SMR-11/18 111 3.2. Chern classes Let E be a complex vector bundle over M, with fibre Cm and group Gl(m; det(XIm-2fc П)= I fk^)*m'k 0£k£ m As a consequence of the properties of £7, they are invariant by ad(Gl(m; (C)). By the precedent theorem, there exists for each к (0 ë k s m) a unique closed 2 к-form Yk on M such that P ' (T'k) = M n ) where p : P -* M is the projection, so that we can write: det(Im ‘ г Ь = P*(1 + 7i + • • • + 7m) Theorem 13: The к-th Chern class ck(E) of a complex vector bundle E over M is represented by the closed 2 к-form Yk. Remark: The Pontryagin classes and Euler classes can be obtained in a similar way by taking a real vector bundle over the manifold M. REFERENCES [1] BOTT, R., CHERN, S. S ., Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections, Acta Math. 114 (1965) 71. [2] CARTAN, E., Les groupes d’holonomie des espaces généralisés, Acta Math. 48 (1926) 1. [3] CARTAN, E., L'extension du calcul tensoriel aux géométries non-affines, Ann. Math. (2), 38 (1947) 1. [4] CARTAN, H ., Notions d'algèbre différentielle; application aux groupes de Lie et aux variétés où opère un groupe de Lie, Colloque de topologie de Bruxelles (1950) 15. [5] CARTAN, H ., La transgression dans un groupe de Lie, Colloque de topologie de Bruxelles (1950) 57. [6] CHERN, S.S., Characteristic classes of Hermitian manifolds. Ann. Math. (2) 47 (1946) 85. [7] CHERN, S.S., Differential geometry of fiber bundles, Proc. Int. Congr. Math, (1950) 2^397. [8 ] EHRESMANN, C ., Sur la notion de connexion infinitésim ale dans un espace fibré et sur les espaces à connexion de Cartan. Nachr. Oest. Math. Ges., (1949) 22. [9] EHRESMANN, C ., Les connexions infinitésimales dans un espace fibré différentiable. Colloque de topologie de Bruxelles (1950) 29. [10] HIRZEBRUCH, Topological Methods in Algebraic Geometry, 3rd enlarged edition, Grundlehren der Math. Wissenschaften 131, Springer-Verlag New-York (1966). [11] HUSEMOLLER, Fiber Bundles, McGraw-Hill, New-York (1966). [12] KOBAYASHI, S. , NOMIZU, K ., Foundations of D ifferential Geometry, Interscience tracts _15 1 (1963) 2 (1969). [13] LICHNEROW1CZ, A ., Théorie globale des connexions et des groupes d'holonom ie. Edizioni Cremonese, Roma (1955). [14] LICHNEROWICZ, A ., Géométrie des groupes de transformations, Dunod (1958). [15] MILNOR, J ., Characteristic classes mimeographed Notes, Princeton University, (1964). [16] PHAM MAU QUAN, Introduction à la géométrie des variétés différentiables, Dunod (1969). IAEA-S MR-11/19 INTRODUCTION TO GLOBAL CALCULUS OF VARIATIONS H.I. EL IASS ON Mathematics Institute, University of Warwick, Coventry, Warks, United Kingdom > Abstract INTRODUCTION TO GLOBAL CALCULUS OF VARIATIONS. In this paper, manifolds of maps, critical point theories and variation integrals are treated within the framework of a global calculus of variations. INTRODUCTION We shall be concerned with mappings of one manifold S into another manifold M. Throughout this paper, we shall assume S to be a compact, connected Riemannian manifold of class C" and dimension ni 1, possibly with boundary 3S. M shall denote a complete Riemannian manifold of class C“ and without boundary (having some finite dimension). TS -* S, TM -*■ M shall denote the tangent bundles of S, M and TsS, TXM the tangent spaces at s G S, x G M. Let L (TS, TM) -*SXM denote the C* vector bundle with fibre L (TsS, TXM) over (s, x) G S X M. If f : S -* M is a Ck map, 0 â к S oo, we denote by f* TM the pull-back of the tangent bundle of M by f, which is then a vector bundle (of class Ck) over S with fibre (f*TM)s = Tf,. M over sG S. Sections g in f*TM can also be interpreted as "vector fields in TM along the map of f", i. e. Ç : S-» TM such that ?(s) G Tf,sj M for all s G S, as is usually found to be more convenient if f is a curve (n = dim S= 1). If n= 1 and thus f is a curve, then the derivative 3f = (d/ds)f(s) is a section in f*TM. Moreover, for n>l we can interpret the (first) derivative 9f of f as a section in the vector bundle L(TS, f*TM) -» S, which has L(TsS,Tf^ M) as a fibre over s G S, the derivative at s being a linear map 9f(s) :TsS-* Tf^ M. Now let there be given a continuous function: F : L (TS, TM) -*• E Then for any C1 map f : S-*M we can form the integral: J (f) = J F O f s where we integrate with respect to the Riemannian measure on S. Let h : S M be a given C“ map and denote by C1(S, M^ the set of f in C^S, M) 113 114 ELIASSON with f(s) = h(s) for all s e 8S and f homotopic to h (fixed boundary). Let ft : S -»■ M be a C1 variation of f = f0 keeping the boundary of S fixed, i. e. (t, s) -* ft (s) is a C1 map : [0 ,e )X S -* M with ft(s) = f0(s) (= h(s)) for s £ 3S att = 0. Then with ? (s> = | f ft(s> lt-oe T f(.)M ? is a field along f, called the corresponding infinitesimal variation of f (a variation ft with a given section f in f*TM as its infinitesimal variation could be given by ft(s) = exp (t f (s)), with exp: TM -» M the exponential map of M). The associated variation of J is given by: dJ (f)-? = ± J (ft)|t=0 dj (f) : tft^'TMlo ->• IR is a linear map, the variational derivative of J, and can in fact be interpreted as the derivative of J : (^(S.M)),-* IR. J is said to be stationary at f or f a critical point of J, iff dj(f) = 0. The basic questions in calculus of variations are: I. If J is bounded below, does J assume its minimum? II. Does J have critical points and what is the structure of the set of critical points? Example 1. S is a bounded domain in IRn with a С boundary, M = ]R and J(f) = | / | | Df(x) f dx = i f J ( - J 0 dx S S i= i Example 2. S = [0,1] С IR, M as above and l J(f) = \ J II 9f(s) (I2 9s о Then, given h, the critical points of J are exactly the geodesics in M from h(0) to h(l) and homotopic to h. Here || Эf(s) || is the norm of the tangent 3f(s) to the curve f(s) in the Riemannian metric of M. J(f) is called the energy of the curve f. Example 3. S and M general as above and IAEA -S MR -11/19 115 the energy of f. Here L(TsS, TXM) is given the metric: n 1. MANIFOLDS OF MAPS We shall first endow the space C°(S, M) of continuous maps S -* M with a C" Banach manifold structure by constructing a special atlas. Let exp : TM -* M denote the exponential map for M and expx the restriction of exp to the tangent space TXM at x in M. The derivative of expx at v€TxM is a linear map: D expx (v) : TXM - TexpvM 116 ELIASSON and D expx(0) = identity is a well known property of the exponential map. Thus there is an open neighbourhood U of the zero section (the set of zero vectors) in TM, such that (t , exp) : U -» M X M is a C°° imbedding of U onto an open neighbourhood W of the diagonal in M X M. Here r : TM-> M is the projection. Now let h; S-* M be a C°° map. Then C°(h*TM) is a Banach space with norm: II? II = sup II I (s) II, ?ec°(h*TM) C° ses h*U is an open neighbourhood of the zero section in h*TM and C°(h*U) is open in C°(h*TM) (remember S is compact.1 ). The map exph : h*U - M; eXPh(? ) = e x p ^ ?, ? G Th(s) M is injective on each fibre and induces an injective map C°(exph) : C°(h*U) -* C°(S,M) C°(exph) (?) = expho ? This map is easily seen to be continuous and open and thus a homeomorphism onto its ima'ge in C°(S,M), which consists of all f£C°(S, M), such that (h, f):S-*W CM XM and we shall denote by C°(h*W). In fact, W-*M given by the projection (x,y)-*x is a fibre bundle and identifying a map of f : S-> M with the corresponding section (id, f) in h*W, C°(h*W) simply denotes the space of continuous sections in h*W and the homeormorphism C°(exph) : C°(h*U) - C°(h*W) is induced by the С fibre bundle equivalence: ( V exph) h!;;U - h*W SS Now let hj, h26 C“ (S,M) and suppose C°(h*W) П C°(h| W) = C°(hfW П h|W ) f ф IAEA-SMR-11/19 117 Then each fibre of W12 = h* W О h^W is a non empty open subset of M and C°(exphi)-1(C°(W12)) = C°(U.) with ui = Kj' exPh¡) C hîU The transform ation of coordinates of an f GC°(W12), f " exPhj°? exPh2or> is then induced by the diffeomorphism: ф12= K 2 > exPh/^Kj- exPhl):Ur U2 С^ехр,, ^„(^(exp ) = С°(Ф12) In order to show that the collection C°(exph) of charts, with h in C°(S,M), is an atlas, we have to show that С°(Ф12) is a C” diffeomorphism. We shall do this in a more general setting using the methods in Ref. [3]. Let VB(S) denote the category of C” vector bundles over S. A section functor Г assigns to each EG VB(S) a Banachable space Г (E) containing C°°(E) as a dense linear subspace and each morphism A6C"(L(E,F)) E,F£ VB(N), the morphism A‘ g in ЦГ(Е), F(F)). Thus Г(Е) is isomorph to the completion of С" (E) in some norm || • || (which is determined only up to equivalence of norm s) and A -1 r S const H € Hr , (A- Ç) (s) = A(s) • ?(s) Let E E VB(N) and choose some Riemannian m etric for E. Then we . p oo define for |G C (E): Il € H с o = SUP II € (s) s 118 ELIASSON and obtain the Banach spaces C°(E) and L P(E) by completing C°°(E) in those norms. Changing the metric on E only changes the above norms into equi valent norms and as l|A -|||co S II a ||co II € ||co l|A-?||LpS II A II co II ? llLp we have section functors C° and Lp. If we choose additionally to the metric a Riemannian connection fo rE , we can differentiate a section f in E с ova riant ly k- times to obtain the к-th-order covariant derivative VKf as a section in L k(TS, E) and then define (for к = 1, 2 , ...... , láp Il C llc„ ■ i II ’'с llc. i = О / л .1/р (I (КМ/) i =0 к 1/P (/¿llv'íf) S i =0 к р This gives the section functors С and L Definition. A section functor Г on VB(S) is called a manifold model, iff (1) We have a continuous linear inclusion F(E)CC°(E), allEeVB(S) i. e. III Ilc 0 s const |U||r all?eC ” (E) (2) We have a continuous linear inclusion r(L(E,F))CL(T(E), r(F)), all E,FGVB(S) i. e. Il A- g II § const ||a || ||||| A6C"(L(E,F)) all?ec“(E) IAEA-SMR-11/19 119 (3) Let E, FE VB(S), UCE open with each fibre Us non-empty and Ф: U-* F a C“ fibre preserving map. Then we have an induced continuous map: Г(Ф) : Г(и) -* r(F), Г(Ф)(|)=в»? Remark. Note that (1) implies Г(и)= Г(Е) nc°(U) is open in Г(Е) Lemma 1. With UCE, Ф: U~* F as in (3) and Г a manifold model on VB(S), Г(Ф) is a C“ map and ОГ(Ф) = Г(В2Ф), where D2$: U - L(E, F) is the C” fibre preserving map defined by D2$(?).n = A ®(C+tn)|t=0 = D®,(Ç).n; ? GUS r, G Es, s e s and Ф5 = Ф I us : Us -► Fs Proof: We have: ф» n- ®°l- (d2®«|). (n-f) = e°(n,?M r)-ç) with l в (v, u) = J (ОаФ (v + t(u - v)) - ОаФ^)) dt о в :U'e u '- L (E, F) a C“ fibre preserving map and U' some convex open subset of U. Then II Г(Ф)(п)- Г(ф) (f) - Г(Б2Ф)(?) • (n -f) ||г S const II Г(0) (l,r¡) ||r II Г) - c II г using (2). Then as в (v,v) = 0 and Г(0) is continuous by (3) we have a 6 > 0 for any e> 0, such that С - T) ||г<б=Ф II Г(0) (f, Г)) Hr < € 120 ELIASSON proving DF(4) = T(D2 Ф), which is a continuous map from F(U) into F(L(E,F)) by (3). r(L(E,F)) is continuously included in L(T(E)), T(F)) and thus Г(Ф) is C1. The rest follows by induction using the formula for DF(i), Lemma 2. Letk=0, 1 ,.... and lSp к > — (n = dim S) P Proof: The proof for Ck is easy, using the "Leibniz rule": V(A • f ) = V A ® f + A • VÇ Property (1) for L^, k>n/p, is just the well known Sobolev imbedding theorem and the others follow from that and the Leibniz rule. I should only remark that one proves the continuity of Г(Ф) most easily by first showing that it maps bounded sets of Г(и) to bounded sets in F(F) and then use estimates as in the proof of Lemma 1. Theorem 1. Let Г be a manifold model on VB(S). Then there is a well defined set of maps r(S,M)CC°(S,M), such that r(S,M) is a C“ Banach manifold modelled on the Banach spaces r(h*TM), heC " (S,M), and Г(ехрь) = C:(exph) | r(h*U) : | - exp of as a chart. Proof: We have f(h*U)C C°(h*U) and define r(S ,M ) as the union of the images of C°(exph) : F(h*U )-* C°(S, M) when h runs through C" (S, M). Lemma 1 guarantees that r(S, M) is well defined and change of coordinates is given by a C“ diffeomorphism. The most important manifolds of maps for variation problems are the Sobolev manifolds L P(S,M), k> n /p. We have shown in Ref. [4] how to construct canonical Finsler structures on these manifolds from Riemannian metrics on S and M. The tangent space of L k(S,M) at f is just L£(f*TM) and the LjJ-norm of a section f in f*TM is defined by the same formula as before, now using the metric and connection on f*TM induced by the Riemannian structure on M [3]. Thus we obtain a norm on each tangent space of L^(S,M), which provides LP(S,M) with a Lipschitz Finsler structure and a C*° Riemannian metric if p = 2 [4] . Each component of L^(S,M) becomes a metric space and we have shown it to be complete [4] . Example 4. S is one-dimensional, n= 1, say, either the torus (circle) IR/Z or the unite interval [0. 1] and then in either case parametrized by 0 S s S 1. Then H^S, M) is a C" manifold (H1 = L 2) as !>■§■. For x e H^S, M) we have t xh 1(s , m ) = h V t m ) IAEA-SMR-11/19 121 and H1(SJ M) is a C“ Riemannian Hilbert manifold with the inner product of two tangent vectors f , r) e H1(x*TM) given by: < € , ,> 0 = J <5(8), Г) (s)> ds 0 We have shown [4, 5] that HWfS.M^TM) ЭН°(х^ М ) I 1 H ^S .M ) Э X is a C°° vector bundle with Riemannian metric given by •( , )>Q. Moreover, taking the tangent of curves x in H^S.M) is a C“ section, x - Эх, in this vector bundle. We shall compute this section in local charts. Let h : S — M be a C“ curve. If f G H^h^TM) is the local representative (co-ordinate vector) of x G H1(S, M), we have x = exp оf . Now let К: T2M—TM be the connection map of M, such that the covariant derivative of afield С is computed by V| = K(3f); 3f(s) = TC (s, 1 ) Tf : TS = S X IR- T2M the tangent of ?. Then with т : TM - M, тг: T2M - TM the projections, (т: , Тт, K) : T2M -TM © TM © TM is a C” diffeomorphism [3 ]. Then we obtain: dx = Tx • 1 = T exp oTf • 1 = T exp о (Tj, Тт , K) 1 о (tj , Тт, K) ° 9f = V exp » (?, 0h, Vf ) = (Vj exp о f ) • 9h + (V2 exp » ? ) • Vf Here we have defined the covariant derivative of the exponential map by Vexp = T exp ° (Tj , Tt ,K) 1: TM© TM0TM - TM and used the fact that Vexp (v,u, w) is bilinear in (u, w) on each fibre [3] . 122 ELIASSON M oreover, V2 exp (v) ■ u = D2 exp (v) • u = -J¡- exp (v + tu) 11 = 0 = D expx(v) • u; v , u e T xM Thus V2exp (v) : TXM-* Texpv M is a linear isomorphism for v sufficiently small, in particular for v£Ü C TM, we used earlier in defining the charts. Now Vj exp (v) : TXM-*- TeXpv M is a linear map and we obtain C°° fibre maps: 9 : U - L (TM, TM) в (v) = (V2 exp (v)) 1 Vj exp (v) 0h : h*U - h*TM; 0h(v) = в (v) • 3h Then we obtain Эх = (D2 expo?) • (Vf + 0h° I) This shows that 8h(i) = V? + 0hci is the co-ordinate vector of Эх, since the local trivialization of the vector bundle H(\ h 1(S,M)*TM) is just given by: hV u ) x H°(h*TM) Э ( ? ,n )- (D2expo?)-n Now we easily see that Э is a C” section, as V; H1(h*TM) -* H°(h*TM) is a continuous linear map and H1(hsiiTM) С H°(h*TM) is of class C°° by Lemma 1. Let us also compute the metric on H^H^S, M)*TM) locally. Let heC"(S,M ), x= exp °? with ? e Hx(h*U) as before. Let Y, ZeH°(x*TM)be locally given by f, Ç, i. e. IAEA-S MR-11/19 123 Then we get: l = <(Gh oÇ)-T), Ç> n 0 with G : L(TM, TM) a ( f fibre map, G(v) = D2exp (v)*D2exp(v) Gh = h*G : h*U - L(h*TM, h*TM) Thus H1(Gh) : h V u ) - H2(L(h*TM)h* TM)) is of class C” by Lemma 1 and we have a continuous linear inclusion H1(L (h*TM, h*TM)) С L(H°(h*TM), H°(h*TM)) by: I|a -?||0 s II a ||co ||?||0 s const II a II j ||ç||0 The metric is thus of class C" and the corresponding norm equivalent to the H°-norm in H°(h*TM) for all f eH ^ h 'U ), if we choose U bounded and such that G(v) ёб > 0 for all v £ U . Now it is immediate that the energy function E(x) = I II Эх ||¡ = i <Эх, Sx>0 is a С” function on H^S.M). The derivative of E extends the variational derivative: dE (x) •? = <Эх, Vf >0 If x is a critical point of E, i.e. x E H^S, M) and dE(x) = 0, then x can be shown to be a C°° curve and then (in case of S = [0, 1], we restrict E to the submanifold of curves having fixed initial and end points): dE(x) • I = - 2. CRITICAL POINT THEORIES Our main aim here is to discuss the condition (Condition (C) by Palais and Smale [13] ), which has been shown to be strong enough to make it possible to generalize Morse Theory and Lusternick-Scbnirelman category theory to infinite dimensions. Let X be a Banach manifold of class C2 at least. A norm on a Banach space bundle E over X is a continuous function v-* || v|| on E, such that the restriction to each fibre Ex of E is an admissible norm on the Banach space Ex. Suppose we have a local trivialization of E over a chart of X: E Э 7Г-1 (U) * ф (U) XIE 77 í I X 3 U - Ф ( и ) с в here В, IE are Banach spaces, cp a C2 diffeomorphism and Ф a homeomorphis: with Фх:Ех->-]Е given by Ф(у) = (cp(x), ®x(v)), v e E ,, a linear homeomorphism. Locally the norm function is given by: N(x.Ç) = II Ф'^х, f )|| The norm on E is said to be uniform, iff for any constant k> 1 and x0Gcp(U), we have: i N(x,Ç) S N (x0, ?) S kN(x, f ) for x in some neighbourhood of x0. Since N(x0,|) is equivalent to the norm of IE, this implies that all the norms N(x,C) on IE, parametrized by x in a neighbourhood of x0, are equivalent to the norm of IE. The norm is Said to be locally Lipschitz, iff for any xfl S cp (U), we have: |N(x.S) - N(x0,f)|s c II x - x0H N(x0,Ç) for some constant с > 0 and x in some neighbourhood of x0. This property is invariant against a change in local trivialization, if E is of class C1 and implies uniformity, as then N(x,f) â (с I x -x 0 I +l)N(x0, €) skN (x0,|) N (x, f) ê (1 - c H x - x0 II )N (x0, C) s 1 /к N(x(x0, Ç ) for any k> 1, if II x- Xq|] is small enough. Definition. A Finsler structure on X is a uniform norm on TX. A Finsler manifold is a regular Banach manifold together with a Finsler structure. IAEA-S MR-11/19 125 Let f be a C1 real-valued function on a C1 Finsler manifold X. f satisfies condition (C) of Palais and Smale, iff any sequence of points xm in X, such that f(xm) is bounded and ||df(xm)|| converges to zero for m -’■<», has a con vergent subsequence. Then of course, if X is complete, the subsequence converges to a critical point of f. The main theorem of the generalized Lusternik-Schnirelman theory of critical points (J. T. Schwartz [14] and R. S. Palais [11] ) is: Theorem 2. Let X be a complete C2 Finsler manifold and f : X-* IR a C1 function bounded below and satisfying condition (C). Then f assumes its minimum on each component of X and has at least cat(M) critical points altogether. Here, cat(M) is the Lusternik-Schnirelman catagory of M, i. e. the number of closed subsets of M, contractible in M, needed to cover M. A good description of this theory can be found in Ref. [12] , with proof of the above theorem. We shall now present a method to establish the important condition (C), see Ref. [6] . A Banach manifold X is called weak submanifold of another Banach manifold X0 containing X as a subset, iff for any x0e X0, there is a chart 9 o : U0 -» cp0 (u0 ) С в0 for XQ with xQ e UQ and a Banach space В e B0 with the inclusion being continuous, such that the restriction of cp0 to U = X £ UQ is a chart cp for X: X 0 Э ^ Фо(и о> c B o и и и x d u ! ф (и) с в ф(и) = ф0(и )п в We shall call such a chart a weak chart for X (at x0). It follows that the inclusion of X into X0 is continuous, but X is not necessarily closed in X 0. We shall call a Finsler structure on X locally bounded with respect to Xq, iff for any x0eX Qand constant L, there is a constant c, such that N (x, f ) s с II Ç II holds for the norm function in xe ф (U), || x || < L, ? e В ( || || denoting the norm in B). Now let X, X0 be Banach manifolds with X a weak submanifold of X0. A function f :X-*IR is called weakly proper with respect to X 0, iff any subset ACX, such that f is bounded on A, is relatively compact in X0, f is called locallybounding with respect to XQ, iff for any x0 in X0 and constant K, there is a weak chart for X at x0 and a constant L, such that Il x II < L for all x e ф (U) with f(x) A С1 function f :X->IR is called locally coercive with respect to X Q, iff for any x0e X0 and constant L, there is a weak chart for X at x0 and constants e>0 and > , such that (in local co-ordinates): 1. (Df(y) - Df(x)) • (y - x) â e D y - x II - A || y - x ||0, for all x, y e tp (U) with ||x 0 < L, Il у II < L, or equivalently, if f is of class C2, such that 2. D2f(z) (I, I ) S e II ? II - A II Ç ||0 , for all x e cp(U) with || x || < L and all Ç e B. Here II II , II 10 denote the norms in B, B0. Theorem 3. Let X be a C2 Finsler manifold and f : Х-» E a C1 function. Suppose there is a C2 Banach manifold X0 containing X as a weak submanifold such that the Finsler structure of X is locally bounded with respect to X0, f is weakly proper, locally bounding and locally coercive with respect to X 0. Then f satisfies condition (C). Proof. Let xm be a sequence in X, such that f(xm) is bounded and ||df(xm )|| -*■ 0 for m-^oo Then as f is weakly proper, we can find a subsequence of xm converging in X0 to some x0e X0. Then we choose a weak chart at x0 , such that the members of the subsequence in the domain cp(U) are bounded in В and such that inequality 1 holds. Now, as the Finsler structure is locally bounded with respect to X0, we may assume N(xm , Ç) S с ||||| and then, using 1: + 7 .(Il df(xm)||+ Il df(xe) Il )c H x m- x j Now as xm is a Cauchy sequence in B0 and || xm || is bounded in B, it follows that xm is a Cauchy sequence in В and thus x06cp(U) and xm -+x0 in B. Theorem 4. Let Г be a manifold model on VB(S). Then the manifold T(S,M) from theorem 1 is a weak submanifold of C°(S,M). Moreover, L^(S,M) for к >n/p is a Finsler manifold with the norm given before (in section 1) and the Finsler structure is locally bounded with respect to C°(S,M). Proof. That r(S,M) is a weak submanifold of C°(S, M) follows immediately from the construction of charts for F(S,M) by restricting the charts of C°(S,M). The regularity of the space r(S,M) follows directly from the regularity of C°(S,M), so l £(S,M) is a Finsler manifold. It is not difficult to show that the Finsler structure of LP(S,M) is locally bounded with respect to С °(S,M), using the local formulas in Ref. [4]. We shall restric t ourselves here to H1(S,M),n= 1, using the local formula: IAEA -S MR -11/19 127 N (x,|) = KG ox) - ? ,D 0 + <(Gox)-Vh(x)-f, Vh(x)-O0)^ where Vh(x) • f is the local representative of the covariant derivative of the vector field (V2exp°x) -f and can be computed by similar methods as used in Example 4 (see Ref. [5] ): Vh(x)-?=V?+ (Л1ох)-|+ (Ao.x)-(0h(x), I) with Aj : h*U - L(h*TM, h’fTM) A0: h*U- L2(h*TM, h*TM) fibrepreserving C" maps and 3h(x) the local representative of the tangent Эх (given in Example 4). Now let L> 0 be given and xSHx(h>:!U) with || x ||j S L. Then Il 9h(x) I I | |V x ||0 + ||eh ox||0 S L + C =Lj II v h (^ ) * € ll0 á II l i o + C i H € II o + C 2L i H € II со á Const II f ||a thus N(x, I) S с II f Hj. Example 5. Theorem 5. Let S be either the unite interval [0, 1] or the circle IR/Z and let HX(S, M)c denote the submanifold in H1(S,M) of curves homotopic to a given C“ curve с : S -► M (with fixed initial and end point, if S = [0, 1] ). Then the energy function E : H^S, M)c->-IR from example 4 is weakly proper (with M compact in case S = Ш/ 2£), locally bounding and locally coercive with respect to C°(S,M) and thus satisfies condition (C). Proof. Let E(x) й К for x € A С H'Vs, M)c , then s dM(x(s), x(t)) s j J К Эх(r ) (I dr I t s s Ш I ^ l 2dT^| ^||9x(t)|| dr|^ t t S I s - t I* (2K)i 128 ELIASSON Thus A is an equicontinuous subset of C°(S, M). A(S) = {x (s ) | x E A, sE S } is relatively compact in M, since M is compact in case S = IR¡Ж, and A(S) is bounded in M by the inequality above (taking t = 0) in case S = [0,1]. Then A is relatively compact in C°(S,M) by the Ascoli-Arzela Theorem. Locally, with x = exp °f, we have E h(?) = E(x) = 1 <(G»f ) • 9„(f ), 8h(f)>o * i 6 II ah{€) II о IIah(€)||0- IIv€||0_ II v 5 M ||€ lli-c0 using G(v)ê6>0 and ||v|| S const for v£h*U. Thus E is locally bounding. Now D E h(?)-n = < (G »i)-8h(|), Vn + (D20h oC)-r,>o + I <(D2G=?)(r!,9h(f)), ah(f)>o D 2E h(?)(rj,rj) = <(G°Ç)(Vr) + (D2 0h = S)-r)), Vr¡ + (D2вьЧ ) • n >q + 2 <(D2G .Ç ). (n,3h(|)), Vn + (D2V Ç) -r)>0 + <(G .€)-8h(f), (D26h°!)(n,r|)>0 + i <(D2G °?)0l,r),ah(?)), 8h(|)>o ë 6||V + (D20h°f) • ri||q - const II riling II 3h(?)||0 II Vr)+ (D2eho|) • n||0 - const ( Il 8h (i ) Il 0 + Il 8h(f ) Il 2 ) Il nII 2 0 6 II II2 x II II2 È ~2 b ill - const II ^ Ilco for II f Hj bounded. For Morse Theory see Refs [8, 9] and for its applications on geodesics Refs [5,7,9]. 3. VARIATION INTEGRALS We shall first consider special differential operators on C” (S,M) with values in a C" vector bundle E-* S X M. We have the vector bundles IAEA-SMR-11/19 129 LJ(TS,TM)->- S X M of j-multilinear maps from TsS to Tx М and can, for any multiindex a = (o^ , . . . ,c*r ), 1 S q>v 5 k, form the vector bundle E a = L(Lai(TS, T M ),... b “r(TS, TM); E ) - S X M of r-multilinear maps with values in E. Let к and w be given integers and suppose we have a C" section A a in E a for each multiindex a , with l S a ySk, a = a^ + . . . + c*r S w. Then, for f 6C " (S, M), we can define V ( “i _1 “f 1 \ P(f)(s) =2^ Aa(s,f(s))- (V 9f(s), . . . ,V affsy CL to obtain a C” section P(f) in (id, f)’:'E-> S. We call P a polynomial differential operator of order к and weight w on E-* S X M and denote the set of those by PDk (E). Here VJ9f is the covariant derivative of order j of the section 8f in L(TS, f*TM) [3]. We call P £ P D |k(S XM,B) (E = S X M X ]R- S X M) strongly elliptic, iff there is a constant \ > 0, such that Aa (s,x) • (tok ®. ?, wk®Ç) s A0 Il u II2k II ÇII 2 for a - (k, k) and all s € S, x6M , u £L(TsS,]R), § e TXM, where uk®f is the element of Lk(TsS, TXM) defined by uk® f (vj ,. . . , vk) = u (vj)...... u (v^JÇ. We can consider P 6 PDk (E) as a section in the linear bundle C" (C” (S, M);:;E)-> C* (S, M), with C°°((id, f)*E) as a fibre over f, and we want to extend P to a C“ section in a larger bundle rj(r(S,M)*E) for a suitable manifold model Г and section functor Г-,. Theorem 6. Let PePD f(E), k>n/p, then P extends to a C°° section in the С vector bundle: L¿(L P(S, M )*E )- L k(S, M) If P e PD™ (E) and к >—, к > m S pk, then P extends to a C" section in the C" vector bundle: ^ Lk-m(Lk(S*M>S:'E ^ L k(S' M> For proof we refer to Ref. [4] . Now we have Vm *9 £ PD™ (Lm (TS, TM)) for the covariant derivation of maps S-*M and thus by Theorem 6 it is a C~ section in: Lk-m (Lk(S’ M)* Lm(TS' TM)) Vm49 t i k > * P L P(S,M) 1 s m s к 130 ELLASSON Then e £: L k(S, M) -*• IR m=l S is a well defined function of class Cr, r Theorem 7. Let g £ C “ (S,M) and PGPDkk(S XM, IR) be strongly elliptic. Then J(f) = f P(f) s is a C“ locally bounding and locally coercive function on H^S.M) with respect to C°(S,M)g (or C°(S,M)). Proof. In a chart Hk(exph) : Hk(h*U)-* Hk(S, M), h € C "(N ,M ), h = g in an open neighbourhood of 9S, we can write the local representative of P: Ph :Hk(h*U )- L1(S) IR), Ph(?) = P(expoÇ) as ph(D = ^T (Aa»f). ( v aiç , . . . ,v “ri) a 1 sk, I a I 5 2k. Since P is strongly elliptic we can establish a Gârding inequality of the form: IAEA-S MR-11/19 131 for some constants > > 0, y ü, y y holding for all r) G H^(h*TM) and f G Hg(h*U) with II i ||c(j <6. Then for 6> 0 sufficiently small (X- y062 § Л/2) we get: в ( € , 0 * 5 II ? II2 - С II? II2 for Hill <á ¿ h С For а ф (к,к) we have: I Г “1 “r I X« = I J Ax 0 ? (v C ...- . v 5)1 s We choose py = 2k/av and obtain |а|Д r-|a|/k Xa S const ||i||k ||Ç||C0 S e U \ \ l + C , for llfll _ e aeb 1'9 á в e a + (1-0) e 1_6b; 0 < в < 1, e > О which holdsfor any positive numbers a, b. Above, в = |а | /2к < 1. Then, summing up and choosing the e 's sufficiently small, we obtain Jh(?)--г N i l ! +const, Hell2 <6 c° or J is locally bounding with respect to C°(S,M). Next we compute the derivatives of J h: DJh(?)-r,=J^ ( D 2Aao ?.(n,vaiÇ...... ,A) S a г V «1 a ar + ¿ Aao?(V "r,...... V rf)) u= 1 132 ELIASSON D2J h(Ç)- (rj.rj) = B(Ç,n) + £ B vf¡(í) ( v V v % ) v + l l < 2 k for Ilf ||k SL, ||g|| 0<5. Thus J is locally coercive with respectto C°(S, M) (a more detailed analysis of the estimates can be found in Ref.[4] ). Variation integrals J as in theorem 7 are possibly not weakly proper, in general. This can be illustrated by the following example: Let Г и ь-l „2 n J(f) = / (I V 9f II , к > — , n > 1 s S = M = Tn= IRn/Z n and consider the sequence fm : Tn-* Tn, given by fm (x)=mx, m = 1, 2. . . . Then 9fm (x):IRn->- IRn is just the multiplication by m and Vk_19f = 0 for к i 2 and so J(fm) = 0. dJ(fm) = 0. So J is bounded on the sequence fm, which obviously does not converge even in C°(S,M). Thus, J is not weakly proper on Hk(Tn, Tn), but it could still be weakly proper on Hk(Tn, Tn)g, as all the maps fm are in different homotopy classes. It is,for example, an open and interesting question whether variation integrals like J(f) = J (H Div 9f||2 + X II 9f ||2 ) s are weakly proper on H2(S,M) (see example 6 here and Ref. [4] ), in general for 2 > n/2. We shall now show that the energy integral E¡, defined earlier is weakly proper, using the following two Lemmas proved in Ref. [4] : Lemma 3. Let n < q < oo. Then there is a constant b, depending only on q and the Riemannian manifold S, such that dM(f(s), f(t)) S b d s (s,t)1‘"/q( y ' d 8f f ) s for all f e C“ (S,M) (and thus also in Lj (S, M)) and s, t£S, where dM,ds denote the distance functions in M, S. Lemma 4. Let 1 S p, q S oo and 0 s m S k, then there is a constant с = c (p, q, m , k, S), such that for any E 6 VB(S) with some Riemannian m etric and connection, we have IAEA-S MR-11/19 133 provided (with > if q = oo and then L~ is replaced by Cm ) and the norms are defined as previously in section 1. In particular, with E = f*TM, f G Lk(N, M), we obtain uniform estimates for norms and distances. Thus a Cauchy sequence in the Finsler manifold L k(S,M), k > n /p , is also a Cauchy sequence in C°(S, M), which is complete, so L k(S,M) is easily seen to be a complete Finsler manifold. Theorem 8. The energy function E k:Hk(S,M)g- R, Ek(f) = I ||9f ||2k l k > n /2 , is weakly proper with respect to C°(S, M), provided M is compact in case 3S is empty, and satisfies condition (C). Proof. We choose q such that 0< 1 - n/qS k- n/2 Then by lem m a 4, we have II af II „ s с II af У = с (2E (f))i " HLP " k-l к and then by lem m a 3: d M(f(s), f(t)) 5 b c (2 E k(f))*ds (s,t)1"n/q for all f £ Hk(S,M)s and s,te S . Thus if Ek is bounded on a subset A of Hk(S,M) , then A is an equicontinuous subset of C°(S,M). Now if 3S is not empty, say s0 G 3S, then f (s0) = g(s0) for all f in Hk(S, M)g and (f(s), h(sQ)) á constant, for f G A or A(S) is bounded in M. Thus, in any case, A(S) is a relatively compact subset of M and then A is relatively compact in C°(S, M) by the Ascoli-Arzela Theorem. We have shown Ek to be weakly proper and it is locally bounding and locally coercive by theorem 7. Thus Ek satisfies condition (C) by theorem 3. Corollary. Let J = J- p , PGPD2k(SXM, IR), k > | s strongly elliptic and M compact, if 3S is empty. Suppose J dominates E k(J(f) ë 7 Ek(f) + const, 7 > 0). Then J : Hk(S, M)g->■ IR is weakly proper with respect to C°(S,M) and satisfies condition (C). 134 ELIASSON Example 6 [4]. Let n= dim S be either 2 or 3 and 9S be empty. Let M be compact and with non-positive sectional curvature. Now H2(S,M) is a complete C” Riemannian Hilbert manifold. We con sider the variation integral: 2 2 / (Il Af II + Л û 3f II ), X >0 S Ш Af(s) = Div 9f(s) = \ Vdf(s) * (еАл e. ), withie-} an ortho- normal base for T S. Then (zi s 1-1 J = J p , with P G PD2 (S X M, IR) s strongly elliptic, so J : H 2(S, M) -* IR is of class C°° and locally bounding and locally coercive with respect to C°(S, M) by theorem 7. We have, see Refs [3,4], V Div 9f • v = ^ V 9f • (v, e., e. ) Í = ^ (V 9f (e., v, e. ) + RM » f (9f • v, 9f • e^ 9f- e. - 9f- Rs(v, e. )e. ) i = (pivV8f + ^ R Mof(9f- , 9. f)9.f - 9f Ricg)- v i with RM, Rg the curvature tensors of M, S and Rics the Ricci tensor of S and 9jf = 9f • e ,. Then 2 J(f) = - = - + <9f Rie. , 9f> + X II 9f II 2 S '0 1 о 5 II V 9f II^ - K J II 9f ||4 + (X + XQ)II 9f ||2 s with XQ the minimum of all eigenvalues of Rics at all tangent spaces of S and К the maximum of all sectional curvatures on M. Now, we have assumed K s 0 and then choosing X ê 1 - X0 , we obtain: IAEA -SM R -11/19 135 j(f) S i II 3f ||2 = E2(f) Thus J is weakly proper with respect to C°(S,M) by the last corollary and satisfies condition (C). The critical points of J are maps of class C" by theorem 16 [4] and if f E C“ (S,M), then dJ(f) • 5 = = <ЛД f + RM ° f (Дf, 9. f) 9jf - А Д f , O o Thus dJ(f) = 0 implies, taking! = -Af: 0 = H V Д f ||2 -< R Mof(Af,9.f)9.f, Af>o + X \\A{ ||¡ and since all three terms are non-negative, we must have Af = 0, or f : S-* M is a harmonic map. By theorem 2, J takes its minimum in every component of H2(S,M) in a harmonic map and we have the lower bounds for the number of harmonic maps given there. This improves the results of Eells and Sampson [2] slightly in case n= 2, 3. REFERENCES [1] EELLS, J., Jr., On geometry of function spaces, Symp. Int.Topol.Algebr. (Mexico 1956) (1958) 303. [2] EELLS, J., Jr., SAMPSON, J.H ., Harmonic mappings of Riemannian manifolds. Amer. J. M ath. 86 (1964) 109. [3] ELIASSON, H .I., On the geometry of manifolds of maps, J.Diff.Geom. 1 (1967) 169. [4] ELIASSON, H .I., Variation Integrals in Fibre Bundles, Proc.Symp. Pure Math. 16, Amer.Math.Soc., (1970) 67. [5] ELIASSON, H .I., Moers Theory for closed curves, Symp.Inf.Dim.Topol.Ann. of Math.Studies 69 (1972) 63. [6] ELIASSON, H .I., Condition (C) and geodesics on Sobolev manifolds, Bull.Amer.Math.Soc. 77 6 (1971) 1002. ” [7] GROVE, K., Condition (C) for the energy integral on certain path-spaces and applications to the theory of geodesics, Preprint series 1971/72, No.4, Aarhus Universitet, Denmark. [8] MEYER, W., Kritische Mannigfaltigkeiten in Hilbert — Mannigfaltigkeiten, Math.Ann. 170 (1970) . [9] PALAIS, R .S ., Morse theory on Hilbert manifolds, Topology 2 (1963) 299. [10] PALAIS, R.S., Foundations of Global Non-linear Analysis, Benjamin, New York (1968). [11] PALAIS, R .S ., Lusternik-Schnirelman theory on Banach manifolds, Topology 5 (1966) 115. [12] PALAIS, R.S., Critical point theory and the minimax principle. Proc.Symp.Pure Math. 15, Amer. Math.Soc., 185. [13] PALAIS, R .S ., SMALE, S ., A generalized Morse theory, Bull. Amer. Math. Soc. 70 (1964) 165. [14] SCHWARTZ, J.T ., Generalizing the Lusternik-Schnirelman theory of Critical points, Comm. Pure Appl.Math. 17(i964) 307. [15] SMALE, S., Morse theory and a non-linear generalization of the Dirichlet problem, Ann.Math. (2) 80 (1964) 382. [16] UHLENBECK, K.K., The Calculus of Variations and Global Analysis, Dissertation, Brandéis University, Waltham, Mass. (1968). IAEA-SMR-11/20 ELEMENTARY SURVEY OF PSEUDO-DIFFERENTIAL OPERATORS AND THE WAVE-FRONT SET OF A DISTRIBUTION R.J. ELLIOTT Mathematics Institute, University of W arwick, Coventry, Warks, United Kingdom Abstract ELEMENTARY SURVEY OF PSEUDO-DIFFERENTIAL OPERATORS AND THE WAVE-FRONT SET OF A DISTRIBUTION. Following Hormander and Sato, the author, successively, deals with the theory of operators with constant coefficients, of operators with variable coefficients and pseudo-differential operators and with the theory of the wave-front set. The results and ideas presented in this paper are due to Hormander and Sato; the treatment follows Hormander [2, 3]. 1. OPERATORS WITH CONSTANT COEFFICIENTS We shall use the standard multi-index notation: a = (ai , an) is a n-tuple of non-negative integers. If x £ R m, x = x ^ xg2 ... x“n , and if D = ^-i , ..., -i , D“ = (-i),a| (^ 7 . •••, * where M =“i + “2 ••• + a n • A linear partial differential operator with constant coefficients can then be written P(D) = Y a « D“ |a|^m where aa are complex numbers. If feCo (Rn) = ¿Z>(Rn), its Fourier transform is defined as л - n /2 Г -i'C x, 6/ f (?) = (27t) J e f(x) dx Rn 137 138 ELLIOTT where dx is Lebesgue measure. It is easy to check, using integration by parts, that - ^- (?) = (1Л> Consider the equation P(D) U = f (1.2) for feCo (Rn). If there is a solution u of Eq.(1.2) with a well-defined F o urier transform u(?) then, from a suitable extension of E q.(l.l): P(?)û(?) = f (?) So by Fourier's inversion formula, we have u(x) = (2тг)"п/2 [ е‘<х’£> f(?)/P(?)d? (1.3) However, P may have zeros on Rn; so, to obtain a solution, it may be necessary to deform the path of integration from Rn into Cn. To show that relation (1.3) can always be interpreted we first prove two lemmas. Lemma 1.1. Suppose $eCc(Cn) and Ф(е10 ?) = Ф(?), 6 e R (*) and / ®(?)dX(?) = l CH<) where dX(?) is Lebesgue measure on Cn. Then, for any analytic function F on Cn, J F(Ç) Ф(?) dX (?) = F(0) (1.4) Proof. By Cauchy1 s formula 2тг f(? ei0)d6 = 2ttF(0) So 2тг J Ф(?) F(C ei9 ) dedX (?) = 2n F(0) Cn 0 IAEA-SMR-11/20 139 but J Ф(f) F (f ei0 ) dX (?) = J 4>(?)F(|)dX(|) C n C n and so the result is proved. Denote by Pol(m) the complex vector space of polynomials in n-variables of degree § m. Write Pol°(m) for the space with the origin removed. F or Q ePol(m ), write Q(“>(Ç) = (iD)aQ for any multi-index a; then || q || = £ |q (“! (0)| |aj£ m defines a norm on Pol(m). Lemma 1.2. If Г2 is a neighbourhood of OeCn there is a C" map Ф: Pol0 (m)-> С0”(П) homogeneous of degree zero, such that the range functions satisfy conditions (*) and (**) and for some constant K: ||q || S К |Q(?) |, Q e P o l0 (m) (1.5) and I e supp. $(Q). Proof. For a fixed QePol°(m), there is a 0eRn such that Q(z0) f 0 for |z| = 1 For this Q choose а Ф with support near the circle zв. An appropriately modified Ф satisfies condition (1.5) and conditions (*), (**). Also this Ф can be usedfor all nearby polynomials. The set of functions satisfying conditions (*), (**) is convex so the construction of Ф can be finishedby means of a partition of unity on the set {Q: | |Q 11 =1}. Finally, we can show how to interpret relation (1.3). Theorem 1.3. There exists a continuous map E: Pol°(m) -* &)' (Rn ) such that P(D)E(P) = 6, the Dirac 6-function, for every PePol°(m). Proof. Consider the expression (Ef)(x) = (2ж )'П/2 / d? f e 1 Rn C n (1.6) where Pj is the polynomial Ç - P (Ç + Ç) 140 ELLIOTT Since one derivative of P is non-zero the function P(€) = ^ |p (0°(C)|, ? e R n |a|^m has a positive lower bound. Thus, by lemma 1.2, P is bounded away from zero in the support of the first integrand and so expression (1.6) is well defined for feCci°(Rn). By lemma (1.1), if we differentiate under the integral we have P(D)(Ef) = f and so we have solved Eq.(1.2) for feCÔ(Rn). The map f -» Ef commutes with translations so there is a distribution which we also denote by E such that Ef = E*f. As (P(D)E)* f = f for all fe C 0"°(Rn) we have P(D)E = 6, the Dirac 6-function. If feá”' (Rn), the space of distributions with compact support, the distribution u = E*f is well defined and satisfies Eq.(1.2). Definition 1.4. The distribution E is called a fundamental solution for the operator P. 2. OPERATORS WITH VARIABLE COEFFICIENTS AND PSEUDO-DIFFERENTIAL OPERATORS In the case of constant coefficients above, the results depended essentially on the Fourier transform, which interchanges (see Eq.(l.l)) differentiation and multiplication. However, differentiation and multi plication do not themselves commute so for operators with variable coefficients one approximates a fundamental solution by constructing a parametrix. Using the above notation, consider a differential operator P with variable coefficients P (x ,D )= ^ T a„(x) D“ (2.1) a ^ m in an open set X cRn. We assume aa(x)eCw(X) and suppose P(x, D) = Pm (x, D) + Pm-! (x, D) + ... is a decomposition of P in a sum of homogeneous term s Pj of degree j. Definition 2.1 P is said to be elliptic if for x*X and о f |«Rn Pm (x,«) = £ a«(x)Ç“ f 0 ( 2 . 2 ) |ot[ = m IAEA-SMR-11/20 141 If P is an elliptic operator with constant coefficients we have for some constant K: I? Г s К I P(f) |, If |> К for f GRn or, indeed, f in some narrow cone in Cn containing Rn. Suppose xeC" (Rn ) is 0 for |f I < К and 1 for large |f.|. Apart from integration over a compact set, which contributes an entire analytic term, the funda mental solution constructed in theorem 1.3 is simply Ef(x) = (2тг)'п/2 J e i Rn Again differentiating under the integral gives a distribution E such that Ef = E*f and P(D)E = 6 + R (2.4) A Here R = x - 1 so that ReC”. Therefore, E is 'almost' an inverse for P and is called a 'parametrix' . Now, E = (2ir ) ~n/2 J е1<х' ё> X(f)/P(f)df Rn SO (-x)“E = (2тг)-п/2 J e1^ -^ D“(x(f)/P(f))df The integrand decreases rapidly at infinity, and so as this identity is true for all a we see that E is C“ away from the origin. Definition 2.2 For v e g 1 (Rn) sing. supp. v is the smallest closed set such that v is C" in the complement, i.e. sing. supp. v = n{x: v = E*P(D)v - R*v Outside sing. supp. P(D)v, E*P(D)veC" as E eC ” outside the origin and R*veC" . Therefore, if we have a parametrix for an operator P, we can say: sing. supp. v = sing. supp. Pv. 142 ELLIOTT Returning to elliptic operators on X with variable coefficients, it is, therefore, of interest to try and construct a parametrix. The classical method of E.E. Levi was to 'freeze' the coefficients at xo6X and to try and find a parametrix as a perturbation of the parametrix E 0 of P(x0,D). Thus, E 0 has the form E 0f(x) = (2 7Г ) "n/2 f e i Rn This parametrix is a better approximation to a parametrix of P(x, D) at x0 than elsewhere, so one hopes a better approximation is obtained if P(x0, ?) is replaced by P(x, |). Note that P(x, f)"1 = Pm (x, f)"1 + ... where the dots indicate homogeneous terms of orders -m-1, -m-2, ... respectively. One is, therefore, led to consider operators of the form E„f(x) = (2*)-n/2 J ei w here, as Ç -► °o, |d “ d ® for x in each compact subset К of X. The set of all functions satisfying* (2.7) is denoted by Sm(X X Rn). Definition 2.3 An operator of the form (2.6) with (peSm(X X Rn) is called a pseudo differential operator of order m with symbol sing. supp. Ev С sing. supp. v (2.8) for all ve S' (X). These properties are all established in Hormander [1]. To find a param etrix for the elliptic operator, we must find a P (x, D + i) We do this by choosing a P (x, D + Ç) ( For j = 0, therefore Definition 2.4 A pseudo-differential operator P of order m and symbol p(x, f) is characteristic at (xJ)eX X (Rn\0) if lim |p(x, tf) |t'm = 0 t-» + “ The characteristic points of P form a closed cone in X X (Rn\0). Let us write char P for the set of characteristic points of P. If no characteristic exists, we say that P is elliptic and the above construction again then shows that we can find an operator Q of order -m so that QP - 1 = Rl5 and PQ - 1 = Rg both have C” kernels. Thus again for elliptic pseudo-differential operators, we have: sing. supp. v = sing. supp. P v; veá^ (X) 3. THE WAVE-FRONT SET Using the above ideas we wish to close with a few remarks concerning the wave front set of a distribution. This interesting concept was first introduced by Sato [4], in connection with his hyperfunctions. In definition 2.2 we defined the singular support of a distribution v e ^ 1 (X) as: sing. supp. v = n {x; Definition 3.1 The wave-front set of a distribution ve¿2>' (X) is WF(v) = П char A the intersection being over all pseudo-differential operators A such that A ue C "(X ). WF(v) is a closed cone in X X (Rn\0 ). 144 ELLIOTT Theorem 3.2 The projection of WF(v) on X equals sing. supp. v. P roof: Clearly, the projection of WF(v) on X is contained in sing. supp. v. If x is not in the projection of WF(v), then we can find finitely many pseudo differential operators A¡ with AjveC , ArAj well defined and {x}X (Rn\0 ) n (Ochar Aj) = j). Then if A = ¿ / A*Aj we have A v e c " and A is elliptic at x so v is C“ there. In the definition of the wave-front set, we see that it is enough to consider operators of order 0. Indeed, it is enough to consider operators of the form b(D)a(x), where b(i) is a homogeneous function of degree 0 for large IС |. As was proved in Ref. [3], this leads to an equivalent definition, which is quite illuminating: Theorem 3.3 (x0, ! 0) ^ WF(v) if and only if for some neighbourhood of x 0 one can find u e S' (X) equal to v in this neighbourhood and v(Ç) = 0( |ç |'N ) for every N in a conic neighbourhood of §o independent of N. Finally the notion of the wave-front set indicates when we may define the product of two distributions Uj, (Rn). Suppose xeCjfR11), ¡X dx = 1 and put X £ (x) = e"nx(x/e). We should like to define uiu2 as the lim it as e -> 0 of (и 1* Х €)(и 2*Х£ )• In general, the limit does not exist. However, if WF(U1) + WF(U2) = {(x, I j + Ç2 ); (x,Ci)eW F(Ui)}cX X (Rn\0) the limit does exist. Further the limit is then independent of x* Finally, let us repeat that we have only sketched ideas and results due to Horm ander and Sato, the full discussion of which can be found in the references. REFERENCES [1] HORMANDER, L., "Pseudo-differential operators and hypoelliptic equations ", Am. Math. Soc. Symp. Pure Math. 10 (1966); Singular integral operators, 138. [2] HORMANDER, L., "On the existence and regularity of solutions of linear pseudo-differential equations", L’Enseignement Math. (1971)99. [3] HORMANDER, L., "Fourier integral operators. I", Acta Math. 127 (1971) 79. [4] SATO, М., "Regularity of hyperfunction solutions of partial differential equations", Actes Congrès Intern. Math., Nice (1970). IAEA-SMR-11/21 BOUNDARY VALUE PROBLEMS FOR NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS R.J. ELLIOTT Mathematics Institute, University of Warwick, Coventry, Warks, United Kingdom Abstract BOUNDARY VALUE PROBLEMS FOR NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS. The paper deals with differential games and the differential equations by which they are governed; in a special section the relevant boundary value problems are discussed. In the last section, so-called "games of survival” are studied. INTRODUCTION The results below are recent work of Kalton and the author [4, 6], generalizing ideas of Fleming [ 7]. Suppose te [ 0, 1], x e Rm, and Y is a compact metric space. Consider a dynamical system x(t) = f(t, x, y),x(0) = x0)1 where f is continuous and Lipschitz in x, and a quantity P = g(x(l)) + / h(t, x, y)dt. Here h: Rm+1 xY-*R is continuous and g is a real-valued function. Optimal control problems study situations of this kind, and the object is to determine a function y(t) so that P (the "cost") is as small as possible. If Z is another compact metric space, an extension of this situation is to consider a dynamical system x(t) = f(t, x, y, z), and real-valued "pay-off" P(y, z) = g(x(t)) + J h(t, x, y, z)dt. We suppose that there are two controllers or players: who controls y G Y and who is trying to make P as large as possible, and J2 who controls z e z and who is trying to-make P as small as possible. Such a situation is called a "differential game". DIFFERENTIAL GAMES Differential games were first studied in the 1950's in rather a formal way by Isaacs, though his results were not published until 1965 [ 10]. The problem when describing a rigorous mathematical model for a differential game is the information pattern; ideally, at any intermediate time each player should know what the other has done up to that time but should not know what he will do in the future. To make this idea precise Fleming [ 8] approximated the situation by replacing the differential equations by difference equations and considering a partition of the time interval into 2n pieces Ij, j = 1, ... 2n. In the upper game corresponding to this partition J2 plays first on each interval and in the lower game, J2 plays second. These games have values W* and W" and Fleming shows that as n -> 00 these converge to quantities W+ and W \ 145 146 ELLIOTT In his work [ 9] Friedman does not approximate by difference equations but he does consider a partition of [ 0, 1] and corresponding upper and lower games where J2 plays first or second on each Ij. In the limit, Friedman obtains upper and lower values V+ and V' and he is able to show V+ = V" if the opposing control variables are separated in f and h, i. e. if f(t, x, y, z) = fj(t, x, y) + f2(t, x, z) h(t, x, y, z) = hjjt, x, y) + h2(t, x, z) Write for the space of measurable control functions from [0, 1] to Y and ..<^2 f°r the similar space with values in Z. A map a -^l called a strategy for if it is non-anticipating, i. e. if, for any T S 1, Zj(t) = z2(t) a. e. 0 S t É T, then ( a z j (t) = (a z2) (t) a. e. 0 É t â T. A strategy a has a value u(a) = inf P (a z, z) zGjc, and there is then an upper value U = sup u(a). Strategies (3 :. л ^ - » are defined similarly and v(|3) = sup P(y, /Зу) y6^i V = inf v(/3) 8 In Ref. [ 2] Kalton and the author show that if the Isaacs minimax condition holds, i.e . for all t G [ 0, 1 ], x ,p £ R m min max (pf + h) = max min (pf + h) z y y z then all the above values are equal: W' = V' = V = U = V+ = W+. This is a stronger result than that of Ref. [ 9]. DIFFERENTIAL EQUATIONS Now consider a game starting from some intermediate time and position (t, x) GRm+1. Then the game starting at this position will certainly have an (upper) value U(t, x) which for a small time 6 will be related to the value U(t + 6, x(t + &)) by a "dynamic-programming"-type identity: t + 6 U(t, x) = min max J' h dt + U(t + 6, x(t + 6) t If U were differentiable then, formally, we should obtain that U satisfies the "Isaacs-Bellman" equation: LU = + min max (VUf + h) = 0 dt z ' y with the boundary condition U(l, x) = g(x) IAEA-SMR-11/21 147 This equation is highly degenerate and there are no results guaranteeing existence or uniqueness of solutions. However, if "white noise" is introduced into the dynamics then the expectation Ux of the upper value can be shown to satisfy the non-linear parabolic equation X2 V2 + 3U^/3t + min max (VU^f + h) = 0 z y for which solutions are known to exist. As X ->■ 0, can be shown to converge to U. Under Lipschitz conditions on f and h, U is Lipschitz and so is a generalized solution of the Isaacs-Bellman equation, in the sense that it satisfies the equation almost everywhere. BOUNDARY VALUE PROBLEMS In his paper [ 7] Fleming changes his point of view. He considers a non-linear equation L*u = # + G(t,x,Vu) = 0, te [0,1], x e Rm dt together with a terminal boundary condition u(l, x) = g(x), and he then constructs a differential game with dynamics: x(t) 1 + 1 у 12У and pay-off 1 P = g(x (1 ) ) + J h dt 0 where 1 + I у I2 y Here У e Y = I y G R m : I y I S <*]■ z e z = I z e R m : I z f s |3 j- If G is Lipschitz it is easily seen that for suitable a and j3, for p e Rn I p I S a and any t, x: max min (pf + h) = G(t, x, p) У z 148 ELLIOTT Therefore, in this case the lower-value function is a generalized solution of the related Isaacs-Bellman equation, which is just the given equation. Furthermore, as any differential game solution is the limit of the unique solutions of the related parabolic equation the differential game solution is unique. GAMES OF SURVIVAL So far, we have considered fixed-time differential games. Consider a set F с Rm+'L such that F з [ T, to) x Rm for some T. The differential game starts at (t0, x 0) and ends the first time tF that the trajectory enters F, the "terminal set". Suppose the pay-off is then 'f P = g(x(tp)) + J h(t, x, y, z) dt 'o If g = 0 and h = 1, P is just the time to the "capture time" t F and so the game is one of "pursuit-evasion". For general g and h, such games are called "games of survival", and they were studied by Kalton and the author in Ref. [4]. The Isaacs-Bellman equation for a game of survival is more complicated because the boundary condition is now U(t, x) = g(t, x) for (t, x) e 3F Kalton and the author show that if there are C1 - functions and 02 such that L S 0 S L 02 and = в2 - g on 3F then U for the survival game is Lipschitz continuous. In Ref. [ 6] we apply these ideas to non-linear boundary problems of the form: L*u = | ^ + G(t, x, Vu) = 0 with u = g on 3F. If there are C1 functions Ojand в2 such that L*e¿ S 0 S L*02 and = в2 = g on 3F, we construct, as above, a related differential game, which is now a game of survival. By the results of Refs [4, 5], the value of this differential game is a solution of the boundary value problem, and again it can be shown that the "differential-game solution" is independent of how a differential game is associated with the equation. REFERENCES [1] ELLIOTT, R.J., KALTON, N.J., Values in differential games, Bull. Amer. Math. Soc. 78 (1972) 427. [2] ELLIOTT, R.J., KALTON, N.J., The existence of value in differential games, Amer. Math. Soc. Memoir 126, Providence R. I. (1972). IAEA-SMR-11/21 149 [3] ELLIOTT, R.J., KALTON, N.J., The existence of value in differential games of pursuit and evasion, to appear in J. Diff. Equations. [4] ELLIOTT, R.J., KALTON, N.J., Cauchy problems and games of survival, to appear. [5] ELLIOTT, R.J., KALTON, N.J., Upper values and stochastic games, to appear. [6] ELLIOTT, R.J., KALTON, N.J., Boundary value problems for non-linear partial differential operators, to appear. [7] FLEMING, W .H ., The Cauchy problem for degenerate parabolic equations, J. M ath. Mech. 13 (1964) 987. [8] FLEMING, W.H., The convergence problem for differential games-II, Ann. Math. Study 52 (1969) 195. [9] FRIEDMAN, A., Differential Games, Wiley-Interscience, New York-London (1971). [10] ISAACS, R. , Differential Games, John Wiley, New York-London (1965). IAEA -SM R -11/22 GAUSSIAN MEASURES ON BANACH SPACES AND MANIFOLDS K.D. ELWORTHY Department of Mathematics, University of Warwick, Coventry, Warks, United Kingdom Abstract GAUSSIAN MEASURES ON BANACH SPACES AND MANIFOLDS. The paper presents some facts concerning probability Borel measures on infinite-dimensional Banach spaces and manifolds. The subjects discussed are cylinder set measures, weak distributions, radonification; abstract Wiener spaces; cylinder set measures versus abstract Wiener space; transformation of integral formula; abstract Wiener manifolds; divergence theorems and theLaplacian.and Brownian motion on a compact m anifold. This contribution is based on that by Eells and the author in Ref. [6]; however, here the emphasis will be more on the abstract theory. Since the subject is rather novel to most mathematicians it seems worth pointing out that measures on infinite-dimensional spaces are not unnatural objects dreamed up by pure mathematicians but the basic stuff of probability theory, of importance in engineering (e. g. stochastic control theory), and of increasing interest in mathematical physics; recently, its interaction with functional analysis has proved extremely rich [13, 14]. A. GENERALITIES We shall be dealing with measures defined on topological spaces, usually separable and metrisable. Our measures will be assumed to be countably additive, positive measures defined on the Borel field of the space; an introduction can be found in Ref. [3]. If (X,ju) is a space with measure /к and Y is a topological space a measurable function f:X->Y determines a measure f(ju) on Y by f(p)B =/uf'1(B) for each Borel set В in Y. Let L0(X,/u;Y) denote the space of equivalence classes of measurable f:Y-* Y. A metric d on Y makes L°(X,p;Y) into a metric space where convergence is convergence in measure: fn -* f in measure iff for each positive e lim /u{x:d(f (x), f(x)) > e} = 0 n—► °° n If (Y, d) is a complete metric space L°(X,/u;Y) is also complete. Throughout we shall be considering real Banach spaces; H will denote a separable, infinite-dimensional Hilbert space, with a fixed inner product. 151 152 ELWORTHY В. CYLINDER SET MEASURES, WEAK DISTRIBUTIONS, RADONIFICATION For a Banach space E let ^af = j^(E) = {T e L(E, Ft ): F t is finite-dimensional, T onto} i. e. is the set of all continuous linear surjections of E onto some finite- dimensional Banach space. A cylinder set measure on E can then be considered as a family { pT : Tej/} of probability measures цт on FT such that given a commutative diagram we have /us = rST(/uT). Examples: (i) Given a m easure ц on E define ц? on F? ky = T(p). By abuse of language a cylinder set measure obtained this way is said to be a m easure. (ii) On H there is a canonical family { yT}TC 0< s < o o of cylinder set measures. Each 7j is given on a Borel set В с F T by y r (B) exp (" ^ в where n = dimFT, | | is the norm coming from the inner product induced on FT by T, and dx denotes Lebesgue measure corresponding to this inner product. Here s is called the variance parameter. A cylinder set measure {/uT}T on E allows one to discuss the integration of tame functions on E: these are those f:E->-R which factorise where Т е ¿s' and fT is measurable. The integral of f over E can then be defined to be IAEA -SM R -11/22 153 It also gives a finitely additive measure /3 on the subsets {T'1 (В): ТЕУ, В C Fx B o rel} i. e. the ring of cylinder sets of E, by /2(T_1 B) = juT(B). The cylinder set measure comes from the measure /u on E iff ц is a countably additive extension of /л. Another way of integrating tame functions on E is by means of a weak distribution. This is an equivalence class of linear functions а:Е*-Ь°(П,р) of the dual of E into the space of real-valued measurable functions on a probability space (fi,p). If «в :E*-*L (^’jP1) then a is equivalent to a' iff for each finite setjij, . . ., ln} С E* the maps a(jfx)X . . .Xa(jen):i2- Rn and a'UjJX . . .X a'(in): r2'^Rn induce the same measures on Rn. Using such maps it is clear that a weak distribution determines a cylinder set measure. The converse follows from general results on projective system s of m easure (e. g. in Ref. [13]), so the two notions are equivalent. A continuous linear map S : E - > G of Banach spaces sends a cylinder set measuref Mt^t on ® int° a cylinder set measure {S (jlí)T} on G by setting S(ju)T = p ToS: The general problem of radonification is to characterize those S for which {S(p)T} is a measure on G (usually for a class of {juT} on E). This has proved a fruitful source for study by functional analysts [13], [14]; however here we shall only discuss the situation where { /ux} is one of the canonical cylinder set measures on H. C. ABSTRACT WIENER SPACES An abstract Wiener space is a triple (i, H, E) where i: H-* E is a continuous linear injection of H onto a dense subset of a Banach space E such that the induced cylinder set measures { í (ys )t } t on E are measures (it suffices to verify it for s = 1). These were originally defined, by Gross, in terms of the condition (*) in the theorem below. Most of the basic results which follow are also due to G ross [8, 9]. 154 ELWORTHY Theorem (Gross, Kallianpur): A continuous linear injection i:H-*E with dense range determines an abstract Wiener space iff: (*) Given e > 0 there is a finite rank orthogonal projection P0 on H such that for any such projection P orthogonal to Po we have 7 p{xeP(H ): H ix К > e> < 1 - e Proof We will sketch the proof of the "if" part only: this is originally due to Gross but we follow Kallianpur's method. A proof of the converse is given in Ref. [10]. Let ¿^"denote the net of finite dimensional orthogonal projections on H and let a: H*-» L°(f2, p) be a representative of the canonical weak distribution on H corresponding to { yt } . For each P in SMve have a measurable map Фр E defined as follows: P:H-^P(H) can be written as П P(x) = ^ (x) e. i= 1 where е 1л . . . , en is an orthonomal basis in P(H) and íj e H*, Set Qp(u) = i a(H; )(co) e¡^. The net ¿“"converges to I, the identity on H. i= 1 For sufficiently large P, Q in ^w e may write P = P 0 + P 1;, Q = Qo+Qi- Condition (*) is then easily seen to be precisely what is needed to show {Qp}pe grC P ° (Q, p; E) is a Cauchy net in m easure. Thus {QP }peSr converges in m easure to some Q e L° (П , p; E). The m easure Q(p) on E is then the required measure which induces {íÍy1)^-}. Examples: (i) Let C 0 (Rn) denote the usual Banach space of continuous cr:[0, 1] -► Rn with a(0) = 0 and let Lo’ (Rn) denote its intersection with the L 2 ,1 functions. Then Lo,:L(Rn) is a Hilbert space with inner product l (ii) When E is a Hilbert space and i:H-*E a continuous linear dense injection then (i, H, E) is an abstract Wiener space iff i is Hilbert-Schmidt [12]. Basic Properties of an abstract Wiener space (i, H, E). 1. Each ys is strictly positive: i. e. each open set in E has positive m easure. 2. The map i is compact. Furthermore its adjoint gives a map j: E*-* H which is also a compact linear injection with dense range and for any T e L(E, Es,s) the map jTi:H-»H is trace class. This was proved by V. Goodman, an unpublished proof is in Kuo's thesis [11], and it depends on the result that || ||E is y 1 square integrable [7]. Thus we have a map tr: L(E, E*)->R, trT being defined as the trace of jTi. Moreover, if LW(E) denotes the space of those maps T: E-»E of the form T = I+ a, i.e. Tx = x + ija(x), where » e L(E, E*) we have a determinant det: L W(E )----► R defined by det(I+o?) = exptr log(I+a). Loosely detT = detT|H considering t |H: H —H. 3. The measure 7 s is quasi-invariant under translation by an element x of E (i. e. 7 s and its image under translation by x have the same null sets) iffx e i(H ), [22]. 4. (see Ref. [33]). The m easure 7 s is quasi-invariant under the action of an isom orphism T eL(E) iff T(i(H)) с i(H) and the induced map т|Н:Н->Н has polar decomposition T |H = U (l+ e) where U: Н-» H is orthogonal and a: H-> H is Hilbert-Schmidt. 5. A strictly positive measure p on a separable Banach space E comes from an abstract Wiener space (i, H, E) iff p is Gaussian i. e. iff i(p) has a Gaussian distribution on R for each l e E*. This was proved by Sato [12] and by Kuelbs. The proof sketched below was told to me by Stefan; see also Ref. [5] for a general discussion. Method of Proof: Suppose p is a Gaussian measure on E. Immediately, there is a linear map E*-* L°(E, p). Since p is Gaussian it follows easily that this maps into L 2 (E, p). Using the strict positivity of p we obtain a linear map T: E * - L2(E, p) which is injective. Let H denote the closure of T(E*) in L 2 (E, p). This is a Hilbert space and T: E* -*H. The Gaussian character of p shows that T is continuous in the weak-* topologies and hence T is the adjoint i* of some map i: H*-» E; this is the only tricky part in the proof and is not needed if E is assumed reflexive. It is easy to check that i induces the measure p as required. The result is easily extended to Gaussian measures which are not strictly positive and the abstract Wiener space structure can be shown to be essentially unique. 156 ELWORTHY 6 . The image i(H) is a Borel set in E and has 7 s measure zero for each s. (See example (iii) in section D below. ) D. CYLINDER SET MEASURES VERSUS ABSTRACT WIENER SPACE From the definition of an abstract Wiener space (AWS) it is clear that it is the canonical cylinder set measure on H which controls its measure theoretic properties. This is emphasized by the result of Gross [9] that if (i, H, E) is an abstract Wiener space there is another AWS (i1, H, E 1) with a factorization i HE E' where k: E '^E is compact. Thus E is just one of many layers of clothes which make the cylinder set measure into a more familiar-looking object for mathematicians. In this sense, the best results will involve E as little as possible and hypotheses such as smoothness with respect to E are rather unnatural and may not be expected to be obtained using measure- theoretic constructions e. g. smoothing operators [19]. The standard method of working with cylinder set measures is to notice that since we can integrate tame functions we should therefore be able to integrate suitable limits of tame functions [8]. A simplified example of this can be described as follows. Choose some AWS (i, H, E) and an orthonomal basis {ei} " =1 for H lying in j(E*). The projections Pn of H onto Span { ei, . . ., e n} extend to projections on E which may be considered as maps Pn : E -* H. Given f: H -> R define fn: E^R by fn = f°Pn . Possibly, {f n} may converge in measure to a measurable f:E^ R, defined almost everywhere. If so, we can define the integral of f over H to be the integral of T over E. Let £ denote the linear space of functions f on H for which such an f exists. We then have a linear map a: /->• L (E, 7 s), a(f) = f ; (it turns out that the choice of s is irrelevant). In some sense f is an extension of f to a function defined almost everywhere on E: but since i(H) has m easure zero in E the notion of 'extension' has to be given a more formal meaning. Examples: (i) It is easy to check that H*C-/ so a linear function on H determines a measurable function on E. Moreover a | H*: H*-> L° (E, 7 s) is a represen ta tive of the weak distribution on H corresponding to the cylinder set measure (ii) For classical Wiener space consider f: L0‘2 1 (R)-*R given by 1 0 IAEA -SM R -11/22 157 for some fixed g eL 2. This could also be written as a Stieltjes integral l f(o-) = J g(t) da (t) о But since almost all functions in C 0(R) have unbounded variation f still does not have a classical meaning almost everywhere on C0 (R). However, f e / and a meaning can, and is, assigned to it as a measurable function on C0 (R); see, e.g. Ref. [16]. This is really just a special case of example (i), but for more compli cated integrands g(t, a(t)) we are led to slightly different methods of extension: the so-called stochastic integrals. There is a whole calculus, developed by Ito [17] and with various modifications [18], to deal with such situations. This highly sophisticated mathematics is much used by engineers, for example in the control of modern chemical plants. (iii) [8 ] Define f: H-> R by f(x) = e"lx^2. Then f fnM d 7 S(x)= / e |x|Z e 2s dx T Rn = (2 s + l ) " n/2 Thus fn -* 0 in L 1 (E, 7 s). Hence, feS^and f is identically zero although f was strictly positive. It follows in particular that the map a is not injective. This example can also be used to show that i(H) has 7 s measure zero by noting that, since fn -» 0 in measure, some subsequence of { fn}must converge to zero almost everywhere [1]. However, fn -> f pointwise on H. Example (iii) shows that one of the main problems in this approach is to characterize a class^ of functions such thato'l^', or some modifica tion of a \ÿ , becomes injective. For further discussion see [15, 8 ]. For measures on manifolds there are more obvious problems as well and it looks as if the Wiener manifold theory described below, which works so well in the abstract, should be considered only as a first approximation which hopefully gives some geometrical intuition to what can actually happen (see section H). Possibly, its relationship to concrete situations is like that of what Arnold calls the Procrustean bed of the theory of Banach manifolds to concrete analysis, which tends to occur on the less tractible Frechet manifolds of C” functions. E. TRANSFORMATION OF INTEGRAL FORMULA For a fixed abstract Wiener space (i, H, E) and an open set U of E a map f: U-> E will be called a C1 W(I)-map if it has the form f(x) =x+a(x) where o:U -> E* is Cr. (Here, and subsequently, j: E*-> H, together with i: H -> E are used to identify E* and H as subsets of E. ) The following 158 EL WORT H Y is slightly weaker than Kuo's original theorem but additional considerations have allowed us to drop his assumption that E* is separable. Theorem (H-H. Kuo [11]). Let Q: U -* V be a C1 W(I)-diffeomorphism between open subsets of E. Define gs (Q, -): U -» R by (Q, x) = I det DQ(x) | exp -j — - 2 < Q(x) - x, x ) - I Q(x) - x f Then J f(y) dy(y )- J f°Q(x) gs(Q,x)dYS(x) v и for any measurable f: V -* R which makes either side exist. Note : The norm in H is denoted by | | to distinguish it from the norm "["j f]” on E. The notation ( , У refers unambiguously to the pairing of E* with E and to the inner product of H. The theorem can be rephrased to say that Q'1 (-ys | V) % ys |u and the -1 / S \ Radon-Nikodym derivative —^ ^ = gs(Q, “). P. ABSTRACT WIENER MANIFOLDS We define a Cr Abstract Wiener manifold (AWM) over (i, H, E) to be a Cr manifold M modelled on E with an equivalence class of atlases {(Ui.QjHi, Q¿: Uj -»E such that eachQjoQj'1 is a CrW(I)-map. This will also be called a Wr~structure on M. This is a minor modification of Kuo's definition. Since the transition maps Q¡ ° Qj:1 of an AWM restrict to diffeomorphisms on the intersections of their domains with H and with E* the charts {(U¡, Qi)}i restrict to give manifold structures to the subsets MH =u Qi-1 (H), -1 * 1 ME# = U Q¡ (E*) of M; the model spaces being H and E, respectively, i Moreover, (for г й 1), applying the same arguments to the derivatives of the transition maps, we have bundles modelled on H and E*, H(M) -» M, E*(M) - M with E*(M) С H(M) С T(M). Let E;(M) — ► HV(M ).----► T M jx x ix x denote the inclusions of the fibres over a point x of M. Note that H(M) restricted to MH is just TMH and similarly for E*(M) restricted to ME*. Property 2 of section С shows that a Wr structure, r è 1, determ ines a Fredholm structure on M [26, 25], and on M E*, and even a "nuclear structure" on MH. In particular it determ ines an element of KO(M) [26]. For a fixed variance param eter s > 0 and an AWM as above define Sij =g*j : Ui n Uj - GL(R) = R-{0} by g¡j (x) = gs (Qj Q jSQ í (x)) IAEA-SMR-11/22 159 Properties of Radon-Nikodym derivatives immediately imply gy (x)gjk(x) = glk(x) x G Uj П Uj (1 U, Hence the family {g^jj j form the transition functions for a line bundle Ж(М) over M [27]. This will be called the bundle of Wiener densities over M and its sections will be called Wiener densities on M (variance parameter s). Such a density Ç is determined by a family {?¡}; of functions f¡ : U¿ -*■ R such that gy (x) ?j (x) = fj (x) for x £ Uj П Uj Since each gy is positive we can say that 5 is a positive density if each >0. Finally such a positive density Ç determines a strictly positive Borel measure ц (Ç) on M by assigning /i(f)(B)= J d7 s(x) Q¡(B) to a Borel set BcU¡. According to Kuo's theorem, this is independent of the choice of chart containing В and it is easy to see that it does, in fact, extend to a countably additive measure as required. In summary, a Wr structure, r è 1, on a Banach manifold M determines a unique measure class on M for each variance parameter and the positive densities determine the measures. In fact, this is automatic, given a transformation of the integral formula, and does not depend, at all, on the actual form of the Radon-Nikodym derivatives involved (although it is convenient to have them continuous). In finite dimensions, there is a geometrical way of obtaining densities, namely by means of a Riemannian metric [2] and this is particularly con venient because it also induces densities on submanifolds; so, for example, submanifolds of Rn have canonically defined measures on them. In our case slightly more is needed because of the lack of translation invariance of our basic measures 7 s. A Cx Wiener-Riemannian metric G on an abstract Wiener manifold M is defined to be a Riemann metric ^ / x on the bundle H(M) over M such that in a chart (Ui(Qi) the local expression for u,v)x can be written as <^Gxu, GXV/ where Gx e L w(E) for each x in Qj(Ui) and x Gx : Qi(Ui) - L W(E) is Cr. Note that for each x in M the choice of such a G makes the inclusion ix : HX(M) -► TXM into an abstract Wiener space (initially HX(M) was just Hilbertable and the definition of an AWS involves a particular inner product); moreover, the map jx: E* (M) -> Hx (M) is that obtained from the adjoint of ix. In particular, E* (M) becomes identified with the cotangent bundle T*M of M since the Riemannian m etric extends to a global pairing of E*(M) with TM. For a density we shall also need a position field, ZonM. This is a vector field Z: M -> TM such that in a chart (Q¿, U¡ ) the principal part of Z, Z1: Qi(Ut) -» E is a W(I)-map. A pair (G, Z) consisting of a WE metric and 160 ELWORTHY a position field will be called Wiener data on M. For such data and for each chart (Q¡, U¡) of M define P¡ = p ¡ (G, Z): Q.(Uj) - R by p. (x) = I det Glv I exp -j - — 2 This can easily be checked to be the local expression of a positive density ps(G, Z) on M, variance parameter s. Thus, Wiener data on M deter mines a m easure /j s = ns (G, Z) on M for all 0 < s < oo. The densities ps(G, Z) also have a more geometrical description. Wiener data (G, Z) induce measures 7 X on each tangent space T XM, yx being defined as the translate of the abstract Wiener measure on Tx M, determined by (ix, HX(M), TXM), by the element Z(x) of T XM. Over a chart QjfUi), for x in QjfUj) the map v«x+v:TxM-»E sends 7 x to a m easure on E which is equivalent to 7 s. Kuo's theorem gives an expression for the Radon-Nikodym derivative, and its value at x is precisely p- (G, Z)(x). Thus, in some sense the m easures 7 X are 'tangent' to the measures // . Another property of the measures ps (G, Z) is that as s -> 0 they converge, at least in some loose sense, to something concentrated on the zeros of Z; possibly this could have interesting applications (see expample (iii) below). Examples: (i) An open subset of E has a trivial W” structure and can be given the standard Wiener data consisting of the given inner product on H and the vector field Z with Z(x) = x. Then ps is just 7 s itself. (ii) If M is Cr Banach manifold, г й 1, and f: M -► E is a Cr Fredholm map of index zero then f induces a uniquely defined W1 structure of M. This follows from the corresponding result about strong layer structures in Ref. [24]. Moreover, when f is proper it is rather easy to obtain an integral formula for the degree of f using this structure [6 ]. From Ref. [25] (or [26], if r s 3) it follows that if M admits Cr partitions of unity and is parallelisable then it admits at least one Wr structure for each element of KO(M). The relationships between the m easure classes corresponding to these structures is not clear. (iii) If E 0 is a finite-codimensional closed subspace of E there is a canonical splitting E = Eq X E 0 where Eq is the annihilator of E 0 in E*. If H0 = E 0 П H then Eq is also the orthogonal complement of H 0 in H. The inclusion i0 : H0 -* E 0 is easily seen to be an abstract Wiener space. Suppose that M is a Cr submanifold of E of codimension n. Choose E 0 of codimension n and let f: M -* Eo be the restriction to M of the pro jection of E onto E0. Then f is Fredholm of index zero and we may apply (ii) to get a canonically defined Wr structure on M, modelled on (i0, H0 ,E Q). This is essentially independent of the choice of E0. We also have Wiener data (GM, ZM) induced on M: GM just being the naturally induced metric IAEA-SMR-11/22 161 from the inclusion of M into E and ZM(x) being defined as the projection of x onto TXM. Thus, any finite co-dimensional submanifold of an abstract Wiener space has a well defined family { ps} of measures on it. As a concrete example take classical Wiener space L 0’ (Rn) -+ C0(Rn) and let M C C ( (Rn) consist of those paths which end on a smooth sub manifold Yn"p of Rn. In this case, the zeros of ZM are just the critical points of the energy integral on M П Lq’1 (Rn). An alternative procedure for assigning a measure to M is given in Ref. [6]. This turns out to be the m easure with density (27ts)"P//2 exp[ - \ \ x - ZM(x) *] p 1(GM, Z M ); see also Ref. [21] and the formula in the divergence theorem below. G. DIVERGENCE THEOREMS AND THE LAPLACIAN A vector field X on an abstract Wiener manifold M will be called admissible if it factors through a section of E*(M)-> M. Let F: D С MXR -* M be the flow of such a vector field X. It is easy to see that each map F t = F (-,t) is a W(I)-map, i. e. is W(I) in the charts of M. If Ç is a positive density on M with corresponding measure p it follows by Kuo's theorem that F t(p | D П MX t) is absolutely continuous with respect to p. Moreover, we can define the divergence of X with respect to Ç by We can now give the simplest divergence theorem: Proposition: Let X be a completely integrable admissible vector field on an abstract Wiener manifold M, of class C1. Then for any positive Wiener density Ç on M with associated m easure p such that p(M) / Div X d/i = 0 provided DivjX G L 1(M,p). Since mathematicians tend to learn the finite-dimensional divergence theorem as a corollary of the much more complicated Stoke's theorem it seems worthwhile pointing out the essential triviality of the proof: Scheme of proof: Formally: _d_ dÇF (p)) d d(F.t (p)) divE X dp dp dt dp d p t = о M t=0 ■ ж I = 0 = f , J ■ I “ |M| Ft(M) the only problem being to justify the differentiation under the integral sign. 1 62 ELWORTHY For the case of a density arising from Wiener data (G, Z) we can find a particularly nice explicit expression for the divergence. The Riemann metric induced by G on H(M) and thence on TMH has a Levi-Civita connection which can be shown to extend to a connection on the whole of TM with connection map l> : TTM ->■ TM having the local form, [28], over (Qi.Ui), f> (x, Ç, y, rj) = (x, r] + l>i (x)(y, ?)) where F>¡ :Q j(Uj ) - L2(EXE, E*). Let V denote covariant differentiation with respect to this connection. Then if X is C1 a computation shows that Div X(x) = tr V X(x) - — < VZ(X), Z> ь g x where Divs refers to the divergence with respect to ps (G, Z). For standard Wiener data on E this reduces to Div X(x) = tr DX(x) - -< X (x ),x > “ s To have more useful hypotheses for the divergence and to deal with manifolds with boundary we shall need the notion of completeness of W-R metrics. We have shown how M contains a Hilbert manifold MH, but in fact each point x of E is contained in the coset x + H of H and these cosets may equally well be pulled back using the charts of M to obtain Hilbert manifolds M h(x) with inclusions M h(x) M such that each point of M lies in one such 'coset' manifold. A W-R metric G on M makes each of these M H(X) into a Riemannian manifold, and we say G is complete if these are all complete in the usual sense. I do not know whether this is equivalent to the geodesic completeness of the connection induced by G on M. A similar result to the following, but for domains in E, has been obtained by Skorohod [20] and Goodman [19]. Goodman's result has interestingly weaker hypotheses which are particularly relevant to the remarks at the end of section H below. Divergence theorem Let M be an abstract Wiener manifold with boundary 3M (possibly 3M= 0) with C 3 Wiener data (G, Z). Assume G is complete. Let X be an admissible C1 vector field on M such that the map x -* |x(x)|x is in L 1(M,ais(G, Z)). Then j DivsX d,us(G,Z) = - j = = = J M M provided both integrals exist, and where n(x) is the uniquely defined internal norm al to 3M at x with |n(x)|x = 1. The basis of the proof is to write X = X j + X2 where Xj is tangent to ЭМ and X2 is normal to ЭМ and has support in a collar neighbournood of ЭМ. We can modify the simple divergence theorem to show that the integral of the divergence of Xj^ vanishes, and can use Fubini's theorem to evaluate the integral of the divergence of X2. The result extends various classical theorems for 'integration by parts' in classical Wiener space [30]. IAEA-SMR-11/22 163 For a С 1 map f:M -»R on an AWM with a W-R m etric we can define grad f to be the admissible vector field determined by the section df of E*(M) = T(M)*. Under the hypotheses of the divergence theorem but with ЭМ = f> we have the éorollary that Divs and-grad are formally adjoint operators. If f is C2 and we define the Laplacian As of f by Asf = Div grad f it follows that Д5 is formally self-adjoint and negative definite. In parti cular if As f = 0 and f is С2 and also f £ L 2 (M, /u2 ) with | grad f |x e L 2 (M, ц* ) it also follows that f is constant. Note that As f has the expression Asf=trV 2f-(l/s)‘\VZ(grad f), Z)> which for M = E with standard Wiener data reduces to As f(x) = tr D2 f(x) - ^ Df(x)x This Laplacian on E has been studied, in a slightly different context, by Um em ura [23]. In particular, its eigenvalues and eigenspaces can be found: the eigenvectors are Fourier-Hermite polynomials. Apparently, it can also be associated with a harmonic-oscillator Hamiltonian in quantum field theory. The Laplacian Д „ defined by A J = tr D2f has been studied extensively by G ross [22]; see also Ref. [21]. For m ore general results on analysis on infinite-dimensional spaces, see Ref. [35] and its bibliography. A stokes1 theorem using "finite co-dimensional differential forms" is in preparation by R. Ram er (Amsterdam). H. BROWNIAN MOTION ON A COMPACT MANIFOLD; CONCLUDING REMARKS For a compact Riemannian manifold X let CXo (X) denote the Banach manifold of continuous paths a: [0, 1] -*X with a (0) = x0. Let LXJ (X) denote the corresponding Hilbert manifold of L2,1 paths [28]. There is a natural Brownian motion measure w on CXo obtained using the fundamental solution of the heat equation on X, [6]; see also Ref. [32] for the relationships between measures, diffusion processes, and parabolic equations. It is'not known whether or not the manifold Cx admits a natural Wr-structure (or even if it admits one at all). However, there is a diffeomorphism, the Cartan development, &): L2,1(TX() X) -> L ^tX ), [6,29]. This does not extend to a map of C0 (TXo M) nor even to one with domain any of the spaces of Holder continuous functions which have, full classical Wiener measure in C0(TXoX). However, we can proceed as in section D and choose projections Pn which map C0(TXoX) onto the finite dimensional subspaces spanned by the vectors of an orthonomal base of L2,1 (TX[l X) consisting of piecewise linear functions. This way we obtain maps 2&° Pn : C0 (TXo X) - L x' (X) which are easily seen to factorise as HL ц’ \р(Х)) c CUo(P(X)) s I 7Г I 7Г L2ll(X) С С (Р(Х)) 164 ELWORTHY for a suitable choice of Qn (which can be explicitly written in terms of exponential maps), where HL2;1 (P(X)) denotes the space of horizontal L 2,1 paths in the principal bundle P(X) of X starting from a fixed frame u0 in TXo (X), and where ît denotes the natural projections. It follows from a (very non-trivial) result of Gangolli [31] that the maps Qn : C 0(TXo X) ->• Cu (P(X)) converge almost everywhere with respect to classical Wiener measure y1 to some Q: C0 (TXo X) -» CUo (P(X)) and that the measure îrQfy1) is precisely the Brownian motion measure w. This important example gives us a measure w on a Banach manifold CXo (X) which is tantalizingly close to fitting into our abstract theory, but apparently does not. Possibly worse, we also have a measure Qfy1) on the closure in CUo (P(X)) of HL u2'-'(P(X)) and this is perhaps not even a manifold, in general; but, nevertheless, the measure is induced in some sense by the dense Hilbert manifold. It seems, therefore, that a useful aim would be to try to extend the abstract theory and try to find exactly what structure is needed on an infinite-dimensional space to obtain a geometrically defined measure with some Gaussian character, and, in particular, to obtain a better understanding of the geometry of these Brownian processes. REFERENCES Basics: [1] K I N G M A N , J. F.C., TAYLOR, S.J., Introduction to Measure and Probability, Cambridge(1966). [2] LOOMIS, L.H., STERNBERG, S., A dvanced Calculus, Addison-Wesley (1968). [3] P A R T H A S A R A T H Y , K.R., Probability Measures on Metric Spaces, Academic Press (1967). General: [4] DUDLEY, R.M., Sample functions of the Gaussian process, M.I.T. (1972). [5] DUDLEY, R.M., F E L D M A N , J., L e C A M , L., O n seminorms and probabilities, and abstract Wiener spaces, Ann. Math. 93 (1971) 390. [6] EELLS, J., E L W O R T H Y , K.D., Wiener integration on certain manifolds, in "Some Problems in Non-Linear Analysis", Centr. Int. Mat. Est. 4 (1970) 67. EELLS, J., Integration on Banach manifolds, Proc. 13th Biennial Seminar Can. Math. Congress, Halifax (1971). [7] FERNIQUE, M.X., Intégrabilité des vecteurs gaussiens, C.R. Acad. Sci. Paris, Ser. A, 270 (June 1970) ' 1698. [8] GROSS, L., Measurable functions on Hilbert space, T.A.M.S. 105 (1962) 372. [9] GROSS, L., Abstract Wiener Spaces, Proc. 5th Berkeley Symp. Math. Stat. and Probability 1965/6, 31. [10] KALLIANPUR, G . , Abstract Wiener processes and their reproducing kernel Hilbert spaces, Z. Wahrschein- lichkeitstheorie 17 (1971) 113. [11] KUO , H.-H,, Thesis, Cornell 1970, published as: Integration Theory on infinite-dimensional manifolds, T.A.M.S. 159 (1971) 57. [12] SATO, H . , Gaussian measure on a Banach space and abstract Wiener space, Nagoya Math. J. 36 (1969) 65. (Note the correction in Ref.[5].) Radonification, functional analytic aspects: [13] S C H W A R T Z , L., Applications Radonifiantes, Séminaire d* Analyse de l'Ecole Polytechnique, Paris (1969/70). S C H W A R T Z , L., Applications Radonifiantes, to appear in J. Math. Soc. Japan, in honour of Yosida (1971). [14] See also articles in Studia Math. 38. IAEA -SM R -11/22 165 Weak distributions: FRIEDRICHS, K.O., SHAPIRO, H.N., Integration over Hilbert space and outer extensions, Proc. Nat. Acad. Sci. U S A 43_ (1957) 336. [15] SEGAL, I.E., Algebraic integration Theory, Bull. A m . Math. Soc. 71 (1965). See also Ref.[8] and the works of Segal. Stochastic integration: [16] NELSON, E., Dynamical Systems and Brownian Motion, Mathematical Notes, Princeton University Press (1967). [17] M c K E A N , H.P., Stochastic Integrals, Monographs on Probability and Math. Stats. No.5, Academic Press (1969). [18] M c S H A N E , E.J., Stochastic Differential Equations and models of random processes, to appear in Proc. 6th Berkeley Symp. Mathematical Statistics and Probability. Divergence theorems: [19] G O O D M A N , V., A divergence theorem for Hilbert spaces, T.A.M.S. 164 (1972) 411. [20] S K O R O H O D , A.V., Surface integral and Green* s formula in Hilbert space, Teor. Verojatnost. Mat. Statist. 2 (1970) 172. (In Russian). STENGLE, G., A divergence theorem for Gaussian stochastic process expectations, J. Math. Anal. Appl. 21 (1968) 537. Potential theory: [21] G O O D M A N , V., Harmonic functions on Hilbert space, J. Functional Analysis 10 (1972) 451. [22] GROSS, L., Potential theory on Hilbert space, J. Functional Analysis 1 (1967) 123. [23] U M E M U R A , Y., O n the infinite dimensional Laplacianoperator, J. Math. Kyoto Univ. 4 (1965) 477. Differential topology: [24] EELLS, J., E L W O R T H Y , K.D., Open embeddings of certain Banach manifolds. Ann. Math. 91 (1970) 465. [25] E L W O R T H Y , K.D., Embeddings, isotopy and stability of Banach manifolds, Comp. Math. 24 (1972) 175. [26] E L W O R T H Y , K.D., T R O M B A , A.J., Fredholm maps and differential structures on Banach manifolds. Global Analysis (Proc. Sympos. Pure Math. 15, Berkeley, Calif., 1968), 45-94 A.M.S. (1970). Nicole M O U LIS: Structures de Fredholm sur les variétés Hilbertiennes, Lecture Notes in Math. 259 Springer-Verlag (1972). [27] STEENROD, N., The Topology of Fibre Bundles, Princeton University Press (1951). Differential geometry: [28] ELIASSON, H.I., Geometry of manifolds of maps, J. Diff. Geo m . 1 (1967) 169. [29] KOBAY A S H I , S., Theory of connections, Ann. di Mat. 43 (1957) 119. KOBAY A S H I , S., N O M I Z U , K., Foundations of Differential Geometry, 1, Interscience (1963). Classical Wiener measure: [30] KOVAL'CHIK, I.M., The Wiener integral, Rus. Math. Surveys 18 (1963) 97. See also Refs [13,16,17, 1]. ~ 166 EL WORTHY Brownian motion on finite-dimensional manifolds: [31] GANGOLLI, R., On the construction of certain diffusions on a differentiable manifold, Z. Wahrschein- lichkeitstheorie 2 (1964) 406. [32] NELSON, E., An existence theorem for second-order parabolic equations, T.A.M .S. 88 (1958) 414. See also Ref.[17]. Additional: DALETSKII, Y u.L ., SHNAIDERMAN, Y a .I., Diffusion and quasi-invariant measures on infinite dimensional Lie groups, Functional Anal. Appl. 3 2 (1969) 88. [33] GUICHARDET, A ., Symmetric Hilbert Spaces and Related Topics, Lecture Notes in Math. 261 Springer-Verlag (1972), [34] ITO, K ., The Brownian motion and tensor fields on Riemannian manifold, Proc. Int. Congr. M ath., Stockholm (1963) 536. NEVEU, J., Processus aléatoires gaussiens, Sém de Math. Sup. Montreal (1968) . VENTSEL, A .D ., FREIDLIN, M .I., On small random perturbations of dynamical systems. Russ. Math. Surveys 25 1 (1970) 1. [35] VISHIK, M .I., The parametrix of elliptic operators with infinitely many independent variables, Russ. Math. Surveys 26 (1971) 91. IAEA-SMR-11/23 SHEAF COHOMOLOGY, STRUCTURES ON MANIFOLDS AND VANISHING THEORY M.J. FIELD Mathematics Institute. University of Warwick, Coventry, Warks, United Kingdom Abstract SHEAF COHOMOLOGY, STRUCTURES ON MANIFOLDS AND VANISHING THEORY. The paper deals with finding significant and computable invariants of structures on topological spaces. After a brief review of sheaf cohomology theory some characteristic situations in complex manifold theory are examined. Stress is laid upon finiteness and vanishing theorems. The material is presented in approximately historical order. In this paper, we wish to consider the following type of problem: Suppose we have a topological space M, together with some structure S^on'M (for example, S^might be a differentiable or complex structure or the struc ture of an analytic or algebraic set). How do we study „9"and find significant (and computable) invariants of S'”? This type of problem has antecedents in the study of topology. Here the problem is of the form: Given a topological space M (no structure) find topological invariants of M which can be used to study M and distinguish it from other topological spaces. Again computability (or manageability) is an important requirement. Of great importance in the study of topological spaces and manifolds has been the development of algebraic topology. Here problems about topological spaces are reduced to essentially algebraic problems in the study, for example, of the cohomology ring of a space. Thus the highly systematised and powerful methods of homological algebra can be applied to topological problems. Henri Cartan in the 1950's, following upon earlier work of his and others on algebraic topology and homological algebra made the fundamental observa tion that the mechanisms and techniques of algebraic topology could be applied successfully to the theory of sheaves. Through the work of Cartan and later Serre, it rapidly became clear that the algebraic topology of sheaves, namely sheaf cohomology, was a tool of great power for analysing structures on topological spaces as well as providing a natural generalization of the coho mology theory of topological spaces. In this paper, apart from briefly reviewing the theory of sheaf cohomology, we shall examine some characteristic situations in complex manifold theory to show the applicability of sheaf cohomological techniques to function theo retic, algebraic and differential geometric problems. In addition, we shall emphasize the general importance of finiteness and vanishing theorems in this context. In a paper of this length, many topics in this area will of necessity have to be omitted; in particular, no discussion will be made of problems in the deformation theory of complex structures or of the general 167 168 FIELD problem of cohomological representation of obstruction to extension of complex structures on complex manifolds. To understand the content of the paper, a knowledge of Ref. [8] (these Proceedings) is required. I have also endeavoured to arrange the material in an approximately historical fashion: the material in the last sections will require somewhat greater mathematical background than that in the first few sections. There will also be a tendency to progress from an informal, conceptual formulation of the material to a more formal problematic presentation. In some ways, this will correspond to a progression from a situation where quite a lot is known (Stein manifolds) to a situation where very little is known (an arbitrary complex manifold). 1. REVIEW OF SHEAF COHOMOLOGY The most natural and general way to define the cohomology of a sheaf of commutative rings 5^over an arbitrary topological space M is by the use of a flabby (or fine) resolution of the sheaf. Here, however, we give a definition based on the Cech construction which is valid only for paracompact spaces (or, alternatively, coherent sheaves): Suppose, therefore, that W = {Uj}isI is an open covering of the topological space M. Let s = (iq...... i p) e I p+1. We set Us = U ^ n ...... ^Uip. We define a p cochain of with values in to be a function f, which assigns to every s £ Iptl, a section fs of &\ Us. In a natural way, the set of all p cochains form a commutative ring which we will denote by C p(^/,gr). We define a coboundary operator <5:CP(<|/, gr)------► Ср+1(^ , &) for p = 0,1,. . by the usual formula: p+l (6f). . =У (-i)V/f. . , \ v v i ^ V o " 4 " lp+1) where, f is a p cochain,^denotes omission and r. : r A j . „ . , g r \ ------►Г/'U. . ,gr\ 1 V V-M-'Vi ) ) is just the restriction homomorphism. ("Г" always denotes sections. ) It is then an easy exercise to check that 62 = 0. With these definitions out of the way, we may define the p-th cohomology group of the complex ICP by Hpfc jr ) = Ker 6 : Cp(^, gr)----- ^ Cp + 1 f a g r ) Im ------.. С The fact that 6Z = 0 implies that the denominator of the above expression is indeed a subgroup of the numerator. IAEA-SMR-11/23 169 If T is a refinement of the cover <%/, then it is not difficult to show that we have a natural homomorphism induced from Hq(‘/' S'") to H4( 1. Exactness. If 0 ------> • ---»- 0 is an exact sequence of sheaves over M, we have an associated longexact sequence of cohomology groups : . . . . ------H\эг2) - Í - * H4( ^ ) ----- H4+1( - U . Н Ч+1(Щ) ------...... 2. H°(5<; = Г (M, (sections of OF). 3. Naturality. If f : Щ ------»■ Щ is a sheaf homomorphism, we have an induced map on cohomology groups f:..: Thus the maps j and i in the exact cohomology sequence are induced from the corresponding maps in the original exact sequence of sheaves. For further details and proofs of the above constructions we refer to Refs [9, 14]. It is worth remarking that we need paracompactness of M to prove exactness of the long exact sequence of cohomology if we define sheaf cohomology via the Cech construction. It should be noted that property 2 above gives an immediate function theoretic description of H° for sheaves of functions. For example, H°(M, & u ), the set of holomorphic functions on M. 2. STEIN MANIFOLDS The first triumphant applications of sheaf cohomological methods to problems in complex manifolds were the celebrated theorems A and В of Cartan. We recall first of all the definition of a Stein manifold: (i) M is holomorphic ally convex (for definition, see Ref. [8] , these Proceedings). (ii) A(M) separates points: Given a, b£M , а ф b, 3f A(M) such that f(a) ф f(b). (iii) Local co-ordinates on M can be defined by globally defined analytic functions [8]. Before stating theorem В we need to define what we mean by a coherent sheaf. We say ^ is a coherent sheaf (of 0 M -modules) on M if ^ has a resolution by free sheaves in some neighbourhood of each point of M. That is, if xeM , there exists a neighbourhood U of x and positive integers p and q such that the following sequence is exact: 04----- 0p----- ^ 1 u --- -0 и и 1 For notations and terminology we refer to Ref. [8] , these Proceedings.! 1 Our definition is deceptive: The weight of coherence lies in properties of the Oka sheaf, 0^, and Oka*s theorem. We refer to Ref. [18] for a fuller exposition. 170 FIELD It follows from this definition and a fundamental theorem of Oka that if ------»-j3?2------► J83 is a sequence of coherent sheaves such that the sequence of stalks л/г x----- x ------* ^ 3 x is exact at some point z 6 M, then it is exact in some neighbourhood of z. This property is characteristic of coherence: "statements or properties about one stalk hold in a neighbourhood of the stalk". This fact enables one often to analyse problems on a single stalk (algebraically) and then get a statement holding in a neighbourhood of the stalk ("topological"). This technique turns out to be extremely powerful in applications. Examples of coherent sheaves are given by:0 M; the sheaf of germs of sections of any holomorphic vector bundle; the ideal or structure sheaf of an analytic set. For proofs and more details about coherence we refer to Refs [14, 18]. We may now state theorem B. Theorem В (H. Cartan): If c®' is a coherent sheaf on a Stein manifold M, then H4(M,_rf] = 0, q â 1. Let us first make one or two observations about this result. Firstly it is an example of a cohomology vanishing theorem. That is, all strictly positive dimensional coherent sheaf cohomology vanishes. This is charac teristic of the type of result that is interesting in the theory (It is saying something about a lack of obstructions to solving various types of problem involving the sheaf). More generally, we might have required some type of finiteness; for example, dim НЧ(У) <». Next we observe that the vanishing of coherent sheaf cohomology implies, often very easily, strong results about the structure on the original manifold. To be precise, we shall prove part of: Proposition: Let M be a complex manifold such that H4 (M,j?/) = 0, qS 1, for all coherent sheaves jé on M. Then M is Stein. (In fact, we shall only need to know that H1 (M, ji) = 0 for all coherent sheaves of ideals on M. ) Proof: The proof of this result is easy, the more so by comparison with the proof of Theorem B, and provides a good example of the advantages of the sheaf cohomological setting. Let us show that M is holomorphically convex. Suppose that К is any compact subset of M. Suppose К is not compact. Choose an infinite subset Q of К with no lim it point in M. Then Q is an alalytic subset of M with an associated ideal sheaf ¿d, say. (Recall that is the sheaf of germs of analytic functions on M vanishing on Q. ) Then is a coherent sheaf of ideals and so Н1(М,^г/) = 0. IAEA-SMR-11/23 171 Associated to the exact sheaf sequence 0 ------»-j/------»-<9M-----Ы ------•’O we have the long exact sequence of cohomology, the first few terms of which are: 0 ------► Н ° Ы ) ------H°( & ) ------► B°(0 ¡já)----- ► h V ) ------►...... M M Hence we obtain the short exact sequence: 0 — ► H 'V )— ► h V m )—- h V m /* o — ►o Now, H°(0M/jsO = T{ 0 j ^ ) is just the set of analytic functions on Q. But since Q is an isolated set of points a function on Q is uniquely determined by assigning an arbitrary complex number to each point of Q. Since H°(0M) ------«• H°(¿?M/V )------*■ 0, it follows that every analytic function on Q is the restriction of an analytic function on M. That is, we may choose an element of A(M), say f, taking an arbitrarily assigned set of values on Q. In particular, we may assume that f is not bounded on Q and hence on К and therefore, by definition of K, on K. Contradiction, since К is compact. The other conditions for a Stein manifold may be equally trivially checked to be true. For the details, we refer to a seminar by Serre in Ref. [29]. Thus far we have seen that knowledge about the cohomology groups of structures can imply strong results about the structures and the underlying topological space. However, again characteristically, the proof of cohomo logy vanishing in Theorem В is hard. For a proof involving the theory of elliptic partial differential equations on non-compact manifolds (proving the 9 sequence exact) together with a non-trivial approximation argument due to Cartan we refer to Ref. [18] . For a proof not involving elliptic theory we refer to Ref. [14] or Ref. [29]. Theorem В is of particular importance as Stein manifolds provide the local models for complex manifolds and the cohomology vanishing gives a straightforward technique for computing sheaf cohomology for arbitrary complex manifolds. To be precise: let ^"be a coherent sheaf on the complex manifold M and let <&'' = {U¡} be an open cover of M by Stein manifolds. Making the observation that an intersection of Stein manifolds is still Stein, theorem В implies that Hq(Us, ¿F) = 0, q§ 1, where Us = U¡ Л ... nu¡ . The theorem of Leray (a direct argument or elementary application of spectral sequences) then gives Hq(«/,^) s Hq(M,án. (For details, see Ref. [9] ) An analogous situation may be shown to hold in the study of analytic sets and algebraic geometry where one has local models (Stein analytic spaces and affine schemata, respectively) which again may be proved to have vanishing sheaf cohomology. Before turning to other examples of vanishing theorems let us note one further application of theorem В to a problem in meromorphic functions, namely the Cousin I problem. 172 FIELD With the notation of Ref. [8], let be the sheaf of germs of meromorphic functions on the Stein manifold M is not a coherent sheaf). Let {Uj} be a cover of M and suppose we are given the principal part of a meromorphic function on each U¡ , m¡G (Uj), such that mj-mj-G A(U¡nUj ). Then there exists a meromorphic function m, defined on the whole of M, such that m-m¡G A(U¡). (This theorem is the several complex variable analogue of the M ittag-Leffler theorem [18]). The proof is trivial. We have the short exact sequence which yields, together with theorem B, the following portion of the long exact sequence of cohomology: H°( 0 н ° с ^ ■ h u(.^ M A(M) But the m¡ define a section ф of 0 M (recall m¡- mj G A(U¡ П Uj )). Hence, there exists m G Г with p(m) = 4. Clearly,m-m; G A(U¡). 3. THE LEVI PROBLEM AND VANISHING OF COHOMOLOGY Let us first of all define a (strongly) pseudo-convex manifold. Thus, suppose M is contained in the complex manifold M as a relatively compact open complex submanifold. M M is said to be strongly pseudo- convex if there exists a differentiable (say C2) function ф :M ------*- R such that: (i) M = {xGM: ф (x)< 0}. (ii) Dф ¡ЭМ^О (This implies 3M is smooth). (iii) The Levi-form of Ф, ддф G С” (T*M ® Т*М) is positive definite when restricted to 3M (Locally ддф is the Hermitian matrix given by d24/dzldzj ). The original examples of strongly pseudo-convex manifolds occurred in problems connected with domains of holomorphy (here M = (Em) when Levi observed that domains of holomorphy with sufficiently smooth boundaries are pseudo-convex and conjectured the converse to be true. This conjecture was proved first by Oka for domains in Œ2 and later for the general case by Bremermann, Norguet and Oka independently Refs [5, 26, 27]. The general problem of the study of convexity conditions on the boundary of a domain and their relation to the function theory of the domain is of great significance in IAEA-SMR-11/23 173 the analysis of boundary value problems in partial differential equations. We refer to R efs[23] , [24], for discussions of the complex pseudo-convex case and to Ref. [19] for a more general discussion valid for the real case. The generalized Levi problem is to prove that a strongly pseudo-convex manifold is holomorphically convex. Grauert in 1958 Ref. [10] proved that if M is strongly pseudo-convex then НЧ(М ,^) is finite-dimensional for q> 0 and & a coherent sheaf. Using this result he then proved that M is holo morphically convex using a straightforward argument of the type used to prove the converse to theorem B. However, the cohomology does not vanish in general and so M is not Stein. A reasonable question is to ask how far away from a Stein manifold is a pseudo-convex manifold and to precisely charac terize the obstructions to its being Stein. We shall briefly review some work of Rossi (see Ref. [28]; this paper also contains an extensive biblio graphy on the Levi problem) which answers these questions in a cohomological framework. The first point to note about a non-compact complex manifold M is that if it has any compact subvarieties it cannot be Stein as we clearly cannot separate points of M lying on the same connected component of a positive dimensional compact subvariety. In the case where M is strongly pseudo- convex it may be shown that the obstruction to M being Stein is represented by the positive-dimensional compact analytic subspaces of M. The precise result is: Theorem: If MCM is stronly pseudo-convex, then (a) If x and у belong to different connected subvarieties of M, there exists an f G A(M) such that f(x) ф f(y). (b) If x does not belong to any strictly positive-dimensional compact sub- variety of M, there exist f j,. . ,fne A(M) which give local co-ordinates at x. (c) M has only finitely many positive-dimensional compact subvarieties. This theorem is deduced from the following cohomological result which gives information about the support of cohomology: Theorem: If MCM is strongly pseudo-convex and J'' is a coherent sheaf on M, then for p>0 there exist f¡ G A(M), 1 S is t such that (a) If I denotes the ideal sheaf generated by fj, . . , ft then the restriction map Hp(M ,3r)------H P(M, © y j I ® jr) is injective. M (b) V = {x e M:fj (x) = 0 Is i s t } consists of finitely many level sets. (Alevelset is of the form: Lc = {z e M:f (z) = с for all f e A(M)} for some c€(C. ) This theorem has the obvious corollary that if M has no compact positive dimensional subvarieties then M is Stein. The main element in the proof of this theorem is the observation that the pseudo-convexivity condition forces the compact subvarieties of M to be bounded away from the boundary of M. 174 FIELD For more details we refer to Ref. [28]. It ought also to be remarked that the use of partial differential equation techniques allows of the possibility of giving precise estimates on the growth of functions on M at the boundary, a feature that is unattainable by purely sheaf theoretic techniques. We also remark that by identifying each connected compact subvariety of M to a point we find that a strongly pseudo-convex manifold M is a proper modification of a Stein analytic space with isolated singular points. Conversely, by Hironaka's resolution of singularities, a Stein analytic space with isolated singularities may be blown up into a pseudo-convex manifold. For more details and references for applications we refer to Ref. [28] . The above description of a strongly pseudo-convex manifold was a rela tive one depending as it did on the space M. We will now look at a slightly different definition of pseudo-convexivity which has the advantage of being intrinsic, though we lose the connections with boundary value problems in partial differential equations. First, by way of motivation, we remark that if we have a domain of holomorphy С (En and define the function p(z) = -log (d(z, 3Í2 )), then the Levi form of p, ЭЭр, is a positive definite Hermitian form. (At least, if the boundary is smooth enough or alternatively we interpret this statement distributionally.) This suggests the following definition: Suppose M is a complex manifold and there exists a function ф : M ------»■ R such that : (i) Mc = {z€M : ф S c} is compact for all c e R. (ii) L( Then we call M a strongly p-pseudo-convex or p-complete manifold. This definition may also be formulated for analytic spaces, but we restrict attention here to manifolds and refer the interested reader to Refs [1,2] for the analytic space case. To avoid confusion with preceding material on the Levi problem we shall use the term p-complete rather than p-pseudo-convex. A domain of holomorphy is thus an example of a 0-complete manifold (proof in Ref. [18] ). Theorem В has generalizations top-complete manifolds. First we recall that a holomorphic p-form Ф on M is just a differential form of type (p,0) such that Ф is holomorphic (and so Ъф = 0). If E is a holomorphic vector bundle on M then we let fip(E) denote the set of holomorphic p-form s on M with values in E. We will also let f2p(E) denote the sheaf of germ s of holo morphic E valued p-forms on M. We have the following vanishing theorem [3, 31] : Theorem: If M is p-complete, then: (i) Hs(M,f2r (E ) ) = 0, s s p + l, r ê 0 (ii) H*(M,f2r (E)) = 0, s S n-p -1. "H*" denotes cohomology with compact supports. IAEA-SMR-11/23 175 The proof is a generalization of the partial differential equation theo retic proof of the exactness of the 9-sequence. This type of result can be of use, for example, when we remove a com plex submanifold of a compact complex manifold and are able to prove that the non-compact manifold resulting is p-complete. We will encounter such an application later. 4. VANISHING THEORY ON COMPACT COMPLEX MANIFOLDS: EMPHASIS ALGEBRAIC In sections 2 and 3 we indicated that vanishing of coherent sheaf coho mology implied strong results about the existence of analytic and meromorphic functions on the underlying complex manifold. The first theorem we have for compact complex manifolds is the finiteness theorem of Cartan-Serre: If .S'is a coherent analytic sheaf on the compact complex manifold M, then H^M ,^-) is finite-dimensional for qêO,Refs [7,14]. Thus if & is the sheaf of sections of a holomorphic vector bundle the theorem says that is finite-dimensional. That is, the set of sections of a holomorphic vector bundle over a compact complex manifold necessarily forms a finite dimen sional vector space. (It is an important problem to be able to compute the dimension of this vector space in terms of topological invariants of the bundle and of M : The most important theorems covering this type of situation are the theorems of the Grothendieck-Hirzebruch-Riemann-Roch type together with their generalisation to complex and differentiable manifolds^the- Atiyah-Singer index theorem. We refer the reader to Ref. [16] , and also to Ref. [4] for details and applications of the index theorem. ) Unfortunately, the theorem of Cartan-Serre, though very useful in certain cases, does not really provide much new information about the sheaf yr or M. However, in the case of algebraic manifolds, Ref. [8], we are able to prove a powerful vanishing theorem, due to Serre, which is analogous to Theorem В for Stein manifolds and characterises algebraic manifolds. The problem of studying the case when M is not algebraic and finding obstructions to M being algebraic or belonging to some other "good" class of manifolds is still open. It is not unlikely that a successful study of more general complex manifolds will depend heavily on a global deformation theory of complex structures. In any case we will not concern ourselves with these problems here but instead restrict ourselves for the rest of this seminar to manifolds of an essentially algebraic type. To motivate the techniques we shall use,we remark that the vanishing of sheaf cohomology on a Stein manifold resulted to some extent from the existence of plenty of analytic functions on M. Now on Pn, whilst there can be no non-trivial analytic functions, nevertheless, as we shall shortly see. there are plenty of meromorphic functions and the form of theorem В for algebraic manifolds will reflect this fact. Let us first of all show how meromorphic functions on complex manifolds correspond to sections of holomorphic line bundles. We must first review the definition and properties of divisors. Let ©''' be the sheaf of germs of invertible analytic functions on M. 0* С 0 and 0* is just the subset of 0 X of germ s of functions which do not vanish at x. We have suppressed the M in 0 M in the above: we shall continue to do this unless there is danger of confusion. 176 FIELD L et.^ ’' be the sheaf of germs of invertible meromorphic functions on M. Here and „^,s =..^- {0} . Neither 0* nor-.-^* are coherent. Let */£?*. eg) is called the sheaf of germ s of divisors on M. A section of Ü? is called a divisor. If d G Г( ÇZ>) and {U¡} is a cover of M, then d is given by a collection { dj}, diG.^*(Ui),such that djd^G 0* (U^nUj ). Notice that if d is a divisor then it defines an analytic subset D of M. In fact, D П U¡ is defined to be the union of the pole and zero set of d¡. Associated to d we also have a line bundle [d] whose transition functions are given by g¡j = djd^1. Another way of seeing this relation is to note that ~ft\M,0*) is isomorphic to the group of holomorphic line bundles on M (group operation tensor product). Then we note that we have the cohomology exact sequence associated to 0 ------► 0 * ------¿f*------*■&)----- ► 0, giving us the sequence Г (*-j on Uj Л Uj. Hence, since g.k = dHdJk, follows that <í>¡/dk= <í>j/dk. In other words, there exists a meromorphic function m on M such that m = «¿’¡/d!' on Uj. Thus sections of Fk correspond in a natural way to meromorphic functions on M. If djGACUj) (that is, d¡ has no poles), it is also clear that m | Uj has poles only on the zero set of dj ; that is,m only has poles on D. It is not, however, a priori clear that a given line bundle has any non trivial sections: take for instance the trivial bundle.' In fact, many line bundles will have no sections at all, other than the zero section. One of the special features of algebraic manifolds is that there exist line bundles with plenty of sections. We now introduce a definition that will enable us to state an important vanishing theorem. Definition: Suppose V is a subset of the complex manifold M. We say V can be blown down to a point if there exists an analytic space X, point x£X and a map f : M ------»-X such that: (i) f(V) = x (ii) f : M -V ------»-X-{x} is an analytic isomorphism. This definition links our work on complex manifolds with that on strongly pseudo-convex manifolds for, if V can be blown down to a point x, it follows (by looking at inverse images of strongly pseudo-convex neighbourhoods of x) that V has a fundamental system of strongly pseudo-convex neighbourhoods. In the next section, we shall be talking mainly about notions of positivity for a vector bundle. Here we introduce the weakest definition of positivity, due to Grauert, Ref. [11] . IAEA-SMR-11/23 177 Definition: If F ------»- M is a holom orphic v e c to r bundle we say: (i) F is weakly negative, if the zero section of F can be blown down to a point. (ii) F is weakly positive if F* is weakly negative. The most important example of a weakly positive line bundle is the hyperplane section bundle of Pn. Associated to the hyperplane H = {(zj,. . , zn + 1) : z 2 = 0} we have a divisor defined by zj = 0. If [H] denotes the corresponding line bundle it may be checked that the transition functions of [H] are given by gy =Zj/Zj. Here we have taken the canonical open covering of Pnby n + 1 open subsets. To show that [H] is weakly positive we prove that [H] is weakly negative. But [H] ' may easily be shown to be the canonical line bundle on Pn induced from the natural principal bundle on Pn : (En+1- {0}----► P n. Then we just note that there is a natural map ф : [H] * - 0-section -> (En+1-{0} which is an isomorphism and as an immediate consequence the zero-section of [H] * m ay be blow n down to a point. If V С Pn is a submanifold note that [H]|V is weakly positive. Thus every algebraic manifold admits a weakly positive line bundle. We have the following weak form of the Kodaira vanishing theorem: T h eo rem : Let M be a compact complex manifold and F a weakly positive line bundle on M. Then^if 7 is a coherent sheaf on M, there exists k(7 , F) G Z+ such that for all k ê k(7 , F): НЧ(М, 7 ® Fk) = 0 fo r q ê 1 The content of the theorem is that, even if Hq(M, 7 ) does not vanish, we can twist the sheaf 7 by tensoring with F k to get vanishing of cohomology. The proof of this result makes essential use of the fact that if D is a strongly pseudo-convex neighbourhood of the zero section of F*, then Hq(D, i^) is finite dimensional for q = 1 and & a coherent sheaf on F*. We lift the sheaf 7 on M to the sheaf 7 = tt*7 ®- 0 * on F* M (again coherent) and show that there exists a natural map: N j : ^ Hq(M ,7 ® F k) ------► Hq(D3y) k = 0 which may be proved to be an injection. Making use of the finite dimen sionality of Hq(D, 3f), for J*" a coherent sheaf on F*, the result follows at once. For further details we refer to Bef. [7] . Before stating the main application of the above theorem we give a definition: 178 FIELD Definition: If F is a holomorphic line bundle & such that for all coherent sheaves & on M, there exists k(F, &) G Z+ such that: H4(M, 3 ? ® F k) = 0, for all kÈk(3sF) and qê 1 then we call the line bundle F, cohomologically positive. Serre proved that the hyperplane section bundle of Pn described above is cohomologically positive,Ref. [29]. In fact, he proved that: Hq(P n, 0 ® [H] n) = 0, qê 1, пй 0. (For proofs see Ref. [29]. ) The importance of the notion of cohomologically positive follows from the fundamental result of Serre which generalizes theorems A and В for Stein manifolds : Theorem: M is algebraic if and only if there exists a cohomogically positive line bundle on M. The implication that cohomological positivity implies M is algebraic is not difficult and in spirit follows that of the converse to Theorem В for Stein manifolds : Let us first show that the condition of cohomological positivity for a line bundle F over M implies that there exist plenty of sections of F k for suffi ciently large k. Suppose, therefore, that a G M. We show that there exists к GZ, such that there exists a section ф of F k with ф(a) =/= 0. If .У denotes the ideal sheaf of the analytic set a and Ф = 0 /j?, we have the exact sequence: 0 ----- »■ j? ------0 ------V ----- ► 0 Clearly, the sheaf Ф is isomorphic the sheaf Ca whose stalk is zero, except at the point a where it is isomorphic to Œ. Tensoring with F k, for sufficiently large k, and using the exact coho mology sequence we obtain the sequence: r ( F k) ----- *F(Fak) ------» 0 since Fk) = 0. Thus, given an arbitrary point zG Fak, there exists a section r of F k such that т (a) = z. Let s0 , . . . , sk be a basis of sections of F N for sufficiently large N. Then we have a map M ----- «-Pk which is defined locally by mapping xl------*■ (s0(x),. . . , sk(x)). It may easily be checked that this map is inde pendent of local representation for the st used. The problem is then to choose IAEA-SMR-11/23 179 N so that this map is an embedding. This is done by expressing the problem in terms of suitable coherent sheaves and using the cohomological positivity. For more details we refer to Ref. [7] . Let us now make a simple application of the existence of a cohomo- logically positive line bundle on Pn to prove Chow's theorem, Ref. [8]. Suppose H is the hyperplane section bundle associated to the divisor Í z-l = 0 } . Let u G Г (Pn, [H]k ), kïO . We have the following Lemma: Any section of [H]k is naturally identified with a homogeneous polynomial of degree к in the coordinates (Zj, . . . , zn+1 ). The proof follows easily. Let Uj = {(zj,. . , zn + 1 ) : z¡ ф 0} , thenUj = € n. Suppose p is a homogeneous polynomial of degree к on We may now easily prove the following theorem of Chow: Theorem: Let A be an analytic subset of Pn. Then there exist homogeneous polynomials f1; . . . ,fm such that A is the common zero locus of the f¡. Proof: We will prove the following: Let be Pn-A. Then there exists a homo geneous polynomial vanishing on A such that f(b) =/= 0. Once this is proved we consider the set of all homogeneous polynomials which vanish on A. Applying the Hilbert basis theorem we choose a finite subset of these poly nomials whose common zero locus is precisely A. Suppose then that denotes the ideal sheaf of b and that F denotes the sheaf of sections of the hyperplane section bundle of P n. We have the exact sheaf sequence 0 ------► ^ ® ^ ® F m------Jf® F m------►j? ®F™----- ►O A b — A — A, b —b For m large enough, the cohomology of J^®_!£®Fm in dimension greater than or equal to one vanishes and so we have the following portion of the exact sequence of cohomology: Г (Pn, J?A ® F m ) ------► Г (Pn, J?A_ b® Ffcm ) ------► 0 Hence there exists f G Г(РП, j^® F m ) such that f(b)=^0. But since С €?n, it follows that Г (Р П, J ^ ® F m ) С r ( P n,F_m). Clearly, Г (Р П, У А ® Fm) 180 FIELD is just the subspace of r(P n,F m) vanishing on A. Since, therefore, f may be regarded as an element ofF(Pn, F m) it follows by the above lemma that f is the homogeneous polynomial whose existence we asserted. In conclusion: In this section, we have given another illustration of how knowledge of the sheaf cohomology of a complex manifold gives important information about the complex structure of the manifold, this time of an algebraic nature. 5. POSITIVITY OF VECTOK BUNDLES AND VANISHING THEORY ON COMPACT COMPLEX MANIFOLDS: EMPHASIS DIFFERENTIAL GEOMETRIC In the remainder of this seminar I wish to survey some work of Griffiths, generalising work of Kodaira, on various notions of positivity for vector bundles. The main reference will be Ref. [12] , though Hartshorne's paper. Ref. [15], should also be consulted for an algebro-geometric presentation. The definition of a weakly positive bundle given in the previous section does not seem to allow of proofs of precise vanishing theorems. Here, we shall introduce three new definitions of positivity for vector bundles, one of a differential geometric nature which allows one to prove precise vanishing theorems, one of a complex analytic nature which expresses the fact that the bundle has 'plenty' of analytic sections and finally a definition of positivity of a topological nature involving Chern classes. Our definitions will follow the paper of Griffiths [12] rather than that of Hartshorne, though with one or two minor differences of detail. In the preceding section, we only defined cohomologically positive line bundles. In generalizations to vector bundles we work, for technical reasons, with symmetric tensor powers of the bundle: Definition: The holomorphic vector bundle E ——»- M is said to be cohomologically positive if, for every coherent sheaf S'- on M, there exists k(E, ¡P) £ Z+ such that: H4(M, .ÿ'® E (k)) = 0, qS 1 and k ï k (E,á?) fk) Here El denotes the к-th symmetric tensor power of the bundle E, some times denoted by O^E. Remark: Using an argument analogous to that sketched in the previous section, where it was shown that a compact manifold admitting a cohomologically positive line bundle is algebraic, it may be shown that a manifold admitting a cohomologically positive vector bundle admits an embedding in a Grassmann manifold (see Ref. [6] ). Since a Grassmann manifold is easily shown to be algebraic, it follows that a manifold admitting a cohomologically positive vector bundle is also algebraic. IAEA-SMR-11/23 181 Definition: E is said to be positive, if there exists an Hermitian metric h for E such that if R£ C” (TM* ® TM* ® Hom(E,E)) is the curvature tensor field associated to h through its connection and R is defined by R = hRe C“ (TM* ® TM* ® E* ®E*) Then R is a positive definite form on decomposable tensors of T ® E. That is, in local co-ordinates, R (C ® r¡ ) = Rq is positive definite in the variables Ç and rj. As ide For details on Hermitian metrics, complex connections, etc. we refer to Ref. [20] . Here we shall just rem ark one or two points about the definition of Hermitian metrics and the notation used above. An Hermitian metric on the bundle E is a smoothly defined set of Hermitian inner products on the vector spaces Ex, one for each xe M. Thus h will be a symmetric linear isom orphism h : E ------»- E* or a section of E* ® E* with the appropriate symmetry and positivity properties (fibrewise). Hom(E,E) denotes the vector bundle whose fibre at x is Ь(Ех,Ех) = Ex ® Ex . Thus Hom(E,E) s E* ® E. We introduce R = hR so as to be able to regard it as a quadratic form on T ® E. TM (or just T), TM denote the holomorphic and antiholomorphic tangent bundle of M, respectively. We remark that if E is positive then ЛГЕ is also positive, where r = dimE, since the induced curvature on ЛГЕ is just trace R ([20] ) which is trivially shown to be positive. Hence a manifold admitting a positive vector bundle admits a positive line bundle and hence is algebraic by standard results (see, for example, Ref. [21] ). We shall show later that this result also follows by proving that positive implies weakly positive. Definition: E —2-*- M is said to be ample if 1. The global sections ofE generate each fibre of E. 2. The natural map - *- Ez ® Tz* is onto. Here Fz is the subset of sections of E vanishing at z. This is the strongest definition of positivity and expresses the fact that, at least up to first order, there are plenty of sections of E. Finally, though we will not make any application here, we give the topological definition of positivity: 182 FIELD Definition: E —3— M is said to be numerically positive if, for all complex analytic subsets W of M and all quotient bundles Q of E| W we have: / P (Cj,. . . , Cs) > 0 w where P is a positive polynomial,Ref. [12], in the Chern classes of Q. Remark: The above definition of numerical positivity is possibly the least well understood of the positivity definitions. Griffiths remarks in Ref. [12] that the definition of a positive polynomial in particular may need to be revised. We give here, however, an example to illustrate the use of Chern classes in the simplest case when E is a line bundle. If E is a line bundle Hom(E,E) = Œ, the trivial bundle, and to say that E is positive is to say that there exists an element w e C"(TM* ® TM*), which is closed under d and is positive definite. We then observe that we have an exact sheaf sequence 0 ------*'°— MM and the following portion of its associated exact cohomology sequences: . . ----- » НМ.0*) H2(M,Z) ------► . . The first Chern class of the line bundle E is then defined as-6(E) and is usually denoted by c^E). Noting that the inclusion j:Z ------*• (C induces a map on cohomology j:H 2(M ,Z )------»■ H2(M, In Ref. [12] , Griffiths proves various relations between the different forms of positivity described above. These relations show, amongst other things, that all types of positivity coincide provided we take a sufficiently high symmetric tensor power of the bundle. In this section, we shall dis cuss and prove some of the relations between the different types of positivity IAEA-SMR-11/23 183 as well as discussing the type of vanishing theorem that can arise in this context to illustrate the techniques used. Suppose that E is a holomorphic vector bundle on M. Associated to E we may construct a fibre bundle P(E) on M, the projective bundle of E. The fibre of P(E) at x is just P(EX), the projective space of Ex. Problems about positive vector bundles E may usually be transferred to questions about positive line bundles by making the observation that we have a canonical line bundle H ------*-P (E*) and that H reflects the positivity properties of E. H is defined by the requirement that н| P(E*)x is the hyper plane section bundle of the projective space P(E*)X. Alternatively, H is the dual of the line bundle associated to the principal bundle E*-M ----»- P(E*). We then have the following results relating positivity properties of H and E: 1. H is cohomologically positive (ample, weakly positive) if and only if E is cohomologically positive (ample, weakly positive). 2. E positive implies H positive. The proof that E positive implies H positive is easily done by a local computation. The equivalence of the weak positivity of E and H follows easily by noting that E*-M з H*-P(E*) and applying the definition. We note also that positivity implies weak positivity. This is best seen by using the metric on E to define the norm function ф = | | 2 : E ------»- R and then noting that a neigh bourhood of fixed radius of the zero section of E is strongly pseudo-convex by computing directly the Levi form L (ф) = 35ф. We refer to Ref. [12] for more details on the proofs of the above implications. Griffiths proves the following generalization of Kodairas's vanishing theorem: Theorem: If E ------»- M is generated by its sections and if F ------► M is a line bundle such that F* ® К ® det E is negative (K is the canonical bundle Лш T*M) then: Hq(M, E_(s) ® F) = 0 q > 0 and s г 0 We shall prove here the classical Kodaira theorem partly: Theorem: If E ------»- M is a negative line bundle then Н Ч(М, £2P(E)) = 0 for p+q< n Corollary (Kodaira vanishing theorem) Hq(M,_E) = 0 for q> 0 if E ®.K* is positive. 184 FIELD The proof of the corollary follows by Serre duality, Refs [21, 30] : НЧ(М,Е) = НП"Ч(М, К ® E*) Proof of theorem: Suppose we have an Hermitian metric h on E and a Kâhler metric on M with Kâhler form H (for example, -i/2©£, where ©E is the curvature tensor field of h on E). We have the inner product defined on E-valued (p,q) forms: (ф,ф)= ^СР,Ч(М,Е) м (Here * is the Hodge star operator and #: E ------*- E* is the conjugate linear isom orphism induced from h. ) Let S> = _ Then 2& is the adjoint of Э with respect to the above inner product: (Ъф,ф) = (ф, &>ф) for феСр,ч"1(М,Е) and фе (? 'q (M,E) We let □ = 3££> + &>d denote the complex Laplace-Beltrami operator, □ : СР,Ч(М, E)----- ►- Cp,q(M, E). Set Hp,q(E) = {ф: П ф = 0} ={ф:дф = Ш>ф = 0} С Cp,q(M,E). Then НР,Ч(Е) is called the space of harmonic (p, q) E-valued form s on M. We now define an operator of great imporatnce in the study of Kâhler manifolds and in Hodge's work in algebraic geometry, Ref. [32] : Let L : Cp,q(M, E ) ----► Cp+1,q+1 (M, E ) be defined by L() =H л . Also set Л= L*, the adjoint of L. Then Л: СР,Ч(М, E ) - ► Cp'1'q"1(M, E). We now need the Nakano inequalities: If ф e Hp,q(E), then i/2 (А(©}л ф,ф) SO ...... A i/2 &лА(ф) s 0 Here © denotes©E, the curvature form of h. These inequalities are valid when M is a Kâhler manifold. We shall not prove them here (we refer to Ref. [21] ),but m erely rem ark that they follow straightforwardly, without local computations, from the following basic equalities for a Kâhler manifold: ЭЛ-ЛЭ=Ш; ЗЛ-ЛЭ = Ш ; (ЭЭ + ЭЭ )ф=@Лф . Suppose then that Е is negative and we have H=_i/2©. The Nakano inequalities, A, imply easily that: ((AL - L A )ф,ф) s 0 В IAEA-S MR-11/23 185 On the other hand,we have the elementary identity (proof in Ref. [32] ): (AL - LA )$ = (n-p-q)...... С В and С imply immediately that: (n— p— q) (ф, ф) S O ...... D The next step uses the only really deep mathematics in the proof. From a theorem of Kodaira (which uses harmonic theory — proof in Ref. [21] ) it follows that Hp,q(E) = Hq(M, Пр (E))...... E E and D imply the result, since (ф,ф) is always positive. Rem arks : For further details on the above proof we refer to Refs [12, 21]. This theorem has many important applications. Kodaira originally used it to prove that Hodge manifolds are algebraic. His proof is given in full in Ref. [21]. Another important application is in the work of Hirzebruch and Kodaira to the study of complex structures on P n, Ref. [17]. Griffiths in Ref. [12] asks the question whether one can prove a precise Kodaira vanishing theorem, for vector bundles, of the form: If E is positive then is Hq(M,E*) = 0 for q s n-r? He proves a special case of this conjecture when r=dimE = 2 and E has a non singular section. Le Potier has since proved this conjecture in full generality. We refer to a seminar by him in these Proceedings. Another result in this direction in Ref. [12] is the following: Suppose E ------»-M is positive and f is a non-singular section of E with zero locus S. If we blow up M along S to obtain M then M-S may be proved to be n-r+1 pseudo-convex (see section 3). Using this fact Griffiths proves that if I denotes the ideal sheaf of S and F is a holomorphic vector bundle on M satisfying a suitable positivity condition then there exists k(F) such that: Hq(M,Ik®F) = 0 for kêk(F) and qSn-r Let us now show how, using the projective bundle P(E*), E positive implies that E is cohomologically positive. The proof will also give the equivalence of cohomological positivity of E and H. Suppose then that H ------P(E*) = P is the canonical line bundle described above. Let ^ be a coherent sheaf on M. We assert that Hq(M ,Iï^ ® &)~ BPiPj^H15 ® 7г*,§г ); the result then follows immediately from the result for line bundles. 1 8 6 FIELD Recall that the p-th derived sheaf Rp (Hk ® ir* & ) of Hk 0 тг*,^ is the sheaf on M associated to the presheaf: U -----► HP(7Г-1 (XJ), H*1 ® v*3r I тг' 1 (U)) We now make use of the fact that n is proper to give the result that the natural map: R^H1' ® г* ЗП (х) ------► HP(Hk ® I tt'^ x )) is an isomorphism (see Ref. [9] ). That is: RP (lí S^\ir'\x)) where ^ is the trivial sheaf over 7г"1(х) = P(E*) з P r_1, and H is the hyperplane section bundle of P(E*). But нр(рг"\н к) = 0 p> 0 ~E¿k) for p = 0 (this is why we use the symmetric product) and it follows that RP(H ® 7rS|i S'”)x = 0,p> 0 « (E(k) ® 5F)x,p = 0 It now follows without difficulty that R°(H k ® тг* S') « E(k) ® S7" An application of the Leray spectral sequence,Ref. [9], gives the result immediately. Finally, as another example of the function-theoretic applications of vanishing cohomology, we prove that if E is cohomologically positive then there exists k> 0 such that E® is ample. P ro of Let zQS M and be the ideal sheaf of z0. Then there exists k (z0) 6 Z+ such that Hq(M, J ® E (k(Zo,) ) = НЧ(У ® E (k(Z<|))) = 0 fo r q> 0 z0 — z0 — IAEA-SMR-11/23 187 From the cohomology exact sequences of 0 z 0 ® — E (k)- E 0 -У®Е 0 Z0 -- — z. z. we obtain Z for kêk(z0). The coherence of the sheaves implies that there exists a neigh bourhood U(z0) of z0 such that Z holds in U(z0) for k = k(z0). We then observe if Z holds for z eU (z0) then Z holds for all positive integral multiples of k(z0). We may then repeat the above process and, using an elementary compactness argument, find к such that E ^ is ample. In a paper of this length we have, of necessity, been forced to omit many topics closely related to the above material. In particular, we have not referred to the theory of deformations of complex structures, Refs [21,22]. Griffiths' paper on the extension problem in complex analysis,Ref. [13], also provides a useful source of examples and problems of a sheaf cohomological type — at least in the formal theory. REFERENCES [1] ANDREOTTI, A., Théorèmes de dépendance algébrique sur les espaces complexes pseudo-concaves, Bull.Soc. Math.France 91 (1963) 1. [2] ANDREOTTI, A., GRAUERT, H., Théorèmes de finitude pour la cohomologie des espaces complexes. Bull. Soc.M ath.France 90_(1962) 193. [3] ANDREOTTI, A., VESENTINI, E., Carleman estimates for the Laplace-Beltrami equation on complex manifolds, I.H.E.S. 2£ 81. [4] ATIYAH, M., SEGAL, G., SINGER, Papers on the Atiyah-Singer Index Theorem in Ann.M ath., in particular I, II, III, 87 (1968) 531. [5] BREMERMANN, H .J., Úber die Âquivalenz der pseudo-konvexen Gebiete und der Holomorphiegebiete im Raum von n komplexen Verà'nderlichen, Math.Ann. 128 (1954) 63. [6] CHILLINGWORTH, D., Smooth manifolds and maps (these Proceedings). [7] DOUADY, A., e tal., Topics in several complex variables, Monographie No. 17 de L' Enseignement Mathématique, Genève (1968). [8] FIELD, M .J., Holomorphic function theory and complex manifolds (these Proceedings). [9] GODEMENT, R., Topologie Algébrique et Théorie des Faisceaux, Hermann, Paris (1964). [10] GRAUERT, H., On Levi's Problem and the embedding of real analytic manifolds, Ann.Math. 68 (1958) 460. [11] GRAUERT, H ., liber Modifikationen und Exzeptionelle Analytische Mengen, Math.Ann. 146 (1962) 331. [12] GRIFFITHS, P.A ., Hermitian differential geometry, Chern classes and positive vector bundles. In 'Global Analysis', Princeton University Press (1969). [13] GRIFFITHS, P.A ., The extension problem in complex analysis II, Am.J. Math. ji8 (1966) 366. 188 FIELD [14] GUNNING, C.R., ROSSI, H., Analytic Functions of Several Complex Variables, Prentice-Hall, Englewood Cliffs, N .J. (1965). [15] HARTSHORNE, R., "Ample vector bundles", I.H.E.S. 29 (1966) 319. [16] HIRZEBRUCH, F., Topological Methods in Algebraic Geometry, Springer (1966); (with appendixes by R .L .E . SCHWARZENBERGER and A . BOREL). [17] HIRZEBRUCH, F., KODAIRA, K., On the complex projective spaces, J. Math. Pures Appl. 36 (1957) 201. [18] HÔRMANDER, L., An Introduction to Complex Analysis in Several Variables, Van Nostrand (1966). [19] HÔRMANDER, L-, Linear Partial Differential Operators, Springer (1964). [20] KOBAYASHI, S., NOMIZU, K., Foundations of Differential Geometry, 1_ and 2 , Wiley Interscience, New York (1963). [21] KODAIRA, K., MORROW, J., Complex manifolds, Holt, Rinehart and Winston Inc., New York (1971). [22] KODAIRA, K., SPENCER D .C ., On deformations of complex analytic structures I and II, Ann. Math. 67 (1958) 328; and also III, Ann. Math. 7^(1960) 43. [23] KOHN, J.J., Harmonic integrals on strongly pseudoconvex manifolds I and II, Ann.Math. 78 (1963) 112. [24] KOHN, J.J., "Boundaries of complex manifolds’’, Proc.Conf.Complex Analysis, Minneapolis, Springer- Verlag (1965). [25] LEWIS, М., An ineffective weakly non-linear unrenormalizable polynomial Lagrangian, I.C .T .P ., Trieste, preprint (1970). [26] NORGUET, F., Sur les domaines d'holomorphie des fonctions uniformes de plusieurs variables complexes (passage du local au global), Bull.Soc. Math. France 82 (1954) 137. [27] ОКА, K., "Domaines pseudoconvexes”, Tôkiku Math.J. 49 (1942) 15. also "Lemme fondamental", J. M ath.Soc.Japan 3_(1951) 204 and 259. [28] ROSSI, H., Strongly pseudoconvex manifolds, Springer Lecture notes No.140 (1972). [29] Seminaire CARTAN, H., 4 (1951-1952), Benjamin, New York (1967); 5 (1953-1954), Benjamin, New York (1967). [30] SERRE, J.P., Un théorème de dualité, Comment.Math.Helv. 2£(1955) 9. [31] VESENTINI, E., On Levi convexivity and cohomology vanishing theorems, Tata Institute, Bombay (1967). [32] WEIL, A., Varieties Kahleriennes, Hermann, Paris (1958). IAEA-SMR-11/24 COMPLEX ANALYSIS ON BANACH SPACES M .J. FIELD Mathematics Institute, University of Warwick, Coventry, Warks, United Kingdom Abstract COMPLEX ANALYSIS ON BANACH SPACES. A brief survey of recent results in complex analysis on Banach spaces is given. In this paper, we wish to give a brief survey of some recent develop ments in the theory of complex analysis on Banach spaces. We shall make no attempt to make a complete survey or provide an extensive bibliography as a survey article by Leopoldo Nachbin should shortly be appearing in the Bulletin of the American Mathematical Society. Recall that a complex-valued function f defined on an open subset U of the complex Banach space E is said to be analytic if f is once continuously differentiable on U with derivative a complex linear map: Df: U - L c (E, Œ) It is well known [1, 12] that this definition is equivalent to any of the following statements: (a) f admits a convergent power series representation at every point of U . (b) f is continuous and f restricted to complex lines is analytic as a map from Œ to Ф. (c) As for (b) except assume that f is continuous at only one point in each connected component of U . (d) As for (b) but assume that f is bounded at only one point of each connected component of U . R em ark s 1. If in property (b) above we completely drop the continuity condition on f we say f is Gâteaux analytic. With the continuity condition f is some times called Frfechet analytic. 2. Using, e.g. property (a) the notion of analytic functions can be defined and studied for more general topological vector spaces, in par ticular, locally convex topological vector spaces. We refer the interested reader to Ref. [ 1] for a development for the non-Banach-space case. The study of analytic functions defined on infinite-dimensional vector spaces is old, having its roots in work of Frëchet and Gâteaux in the 1900's. Interest in the theory in the past decade has been stimulated for several reasons. In the first place, Banach-space complex analysis techniques have proved valuable in the study of essentially finite-dimensional problems. 189 190 FIELD In particular, we mention the work of Douady on the problem of moduli [ 9] (this paper provides the standard reference for the basic theory and defi nitions on complex Banach manifolds). Secondly, work of Eells and Elworthy, following on work of Kuiper, Burghlea, Moulis and others on infinite dimensional differential topology, shows that every separable Hilbert manifold admits a complex structure (in fact, many'. See Ref. [3]). Examples of such manifolds include Sobolev classes of maps from a compact differentiable manifold to another differentiable manifold. Krikorian (Thesis at Cornell) using ideas of Douady has shown that Cr(M,N), r ë 0, admits a natural complex structure when N is a complex manifold. The general interest shown in infinite-dimensional differential topology has also provided a stimulus to develop a corresponding theory for the analytic case. One feature here is that many Banach spaces, in particular C[ 0, 1] (con tinuous functions on the unit interval) do not admit any differentiable bump functions, though they do have, of course, plenty of analytic functions (with non-bounded support) on them. Thus, in some cases it may be more natural to work with analytic functions and techniques than differentiable ones. Having discussed a little of the motivation and history of the subject we now turn to a discussion of some of the problems and the results so far obtained. Noting that the principle of analytic continuation still holds in Banach spaces a very natural problem is to generalize some of the results about holomorphy domains that hold when E is finite-dimensional. However, new problems are faced when one tries to formulate a good definition of "domain of holomorphy" . For example, in finite dimensions a domain of holomorphy is always the domain of existence of an analytic function (see Ref. [ 10], for proof). This is no longer so in infinite dimensions [ 13] . One of the main constructional tools used in finite dimensions — filling up the domain with a countable set of closed and bounded subsets and constructing analytic functions inductively — no longer works for the reason that analytic functions will not, in general, be bounded on a closed and bounded set unless it is compact and one cannot fill up open subdomains of an infinite-dimensional Banach space by a countable set of compact subsets (E is not locally compact). This leads to the study of special classes of analytic functions on a given subdomain £2 of E, for example, the set of analytic functions which are bounded on all closed and bounded subsets of Г2. For this class of functions, results of the Cartan-Thiillen type have been proved [ 7]. A problem, special to the infinite-dimensional case, occurs when we ask how far a domain of holomorphy can be described in terms of its inter sections with finite-dimensional (linear) spaces. For example, ifiicE possesses property X, where X might be: pseudo-convex, domain of existence, domain of holomorphy, etc., then fi n (Dm also possesses property X. The reverse implication is much more difficult, but results have been obtained when fi satisfies some polynomial convexity conditions [19, 21]. Given an arbitrary open subset ficE one can construct its envelope of holomorphy and prove uniqueness in a manner sim ilar to that done in Ref. [ 10] . We refer to Ref. [ 14] for a full treatment. More difficult is the relation between the envelope of holomorphy constructed via sheaves and the spectrum of A(fi). The spectrum of A(fi) was first studied by Coeuré in Ref. [ 5]. I understand that equivalence between the two approaches has now been attained, at least in some cases. lAEA-SMR-11/24 191 In Coeuré's study of A(Q) he is faced with the problem of topologizing A(U) as a topological vector space. The compact open topology appears not to be a suitable topology for studying problems of analytic extension and he constructs a topology suitable for study of analytic extension. Study of the vector space A(S7) leads to intriguing problems in functional analysis associated with the fact that there are several natural topologies that one can put on A(Q). One feature of these topologies is that they generally have the same bornology. We refer to [ 6, 17] for further details. Another interesting area of development has been in work of Gupta [11] and Nachbin on spaces of nuclear analytic maps. Results have been obtained here generalizing work of Martineau [ 16] and Malgrange [15] on convolution operators. A nuclear analytic map is essentially a non-linear generalization of a trace-class operator and is strongly approximatable by analytic maps of finite rank. Other classes of analytic maps have also been studied, in particular by Boland and Dwyker. Hirschowitz's construction for the envelope of holomorphy allows one to construct a nuclear envelope of holomorphy, maximal for analytic continuations of nuclear analytic maps. Probably the most productive area in infinite-dimensional complex analysis has been in the algebraic study of the local rings 0 O(E) and analytic sets. To obtain a satisfactory theory, one has to assume that one's analytic sets are defined by finitely many analytic functions locally (examples by Douady and others show the importance of this restriction: every compact metrizable topological space may be represented as an analytic set if the restriction on finiteness is removed). The problem is then to study the relation between analytic subsets of E and ideals of 0 O (E). Work in this area has been done by Ramis, Ruget, Mazet and others and is well convered in Ramis' monograph [ 20]. Chief among the results are characterizations of the ideals that corre spond to analytic sets defined by finitely many analytic functions, the nullstellensatz for such analytic sets and generalizations of Remmert- Stein's theorem and Chow's theorem to infinitely many variables. Problems outstanding involve finding coherence properties of 0 O (E) and a deeper study of the local ring 0 O(E) with a view to generalizing Oka's normalization theorem . REFERENCES [ 1] BOCHNAK, SICIAK, papers in Studia M athematica on analytic functions on topological vector spaces (1970 -). [ 2 ] BREMERMANN, H. J., Holomorphic functionals and complex convexivity in Banach spaces, Pac. J. Math. 7 (1957) 811. [3] BURGHELEA, D., DUMA, A ., Analytic complex structures on Hilbert manifolds, J. Diff. Geom. 5 (1971) 371. [4] CARTAN, H., Seminaire Bourbaki. [5] COEURE, G., Thesis, Ann. Inst. Fourier, 20 1 (1970) 361. [ 6] DINEEN, S ., Holomorphy types on a Banach space, Studia Math. (1971). [7] DINEEN, S., "The Cartan-Thullen theorem for Banach spaces”, Ann. Sc. Norm. Super. Pisa 24 4(1970) 667 (correction following). [8] DINEEN, S., HIRSCHOWITZ, A ., "Sur le theorem de Levi-Banach", C. R. 272 (1971) 245. [9] DOUADY, A ., "Les problèmes des modules . . Ann. Inst. Fourier 16 1 (1966) 1. [10] FIELD, M .J., Holomorphic function theory and complex manifolds, these Proceedings. 192 FIELD [11] GUPTA, С ., Malgrange theorem for nuclearly entire functions of bounded type on a Banach space, Notas de M athematica, No. 37, Instituto de Mathematica Pura e aplicada, Rio de Janeiro. [12] HILLE, E .# PHILLIPS, R.S., Functional analysis and semi-groups, Amer. Math. Soc. Colloq. Publ. 31 (1957). [ 13] HIRSCHOWITZ, A ., "Remarque sur les ouverts d'holomorphie d'un produit denombrable de droites", Ann. Inst. Fourier, 19 (1969) 219. [14] HIRSCHOWITZ, A ., "Prolongement analytique en dimension infinie", C.R. 270 (1970) 1736. [15] MALGRANGE, B., "Existence et approximation des solutions des équations aux dérivées partielles et des équations des convolutions", Ann. Inst. Fourier £(1955) 271. [16] MARTINEAU, A ., "Sur les fonctionnelles analytiques et la transformation de Fourier-Borel", J. d'Anal. Math. 11 (1963) 1. [17] NACHBIN, L., Topology on Spaces of Holomorphic Mappings, Springer-Verlag (1969). [18] NOVERRAZ, Ph., Thesis, Ann. Inst. Fourier 19^2 (1969) 419. [19] NOVERRAZ, Ph., "Sur la convexité fonctionnelle dans les espaces de Banach", C.R. 272 (1971) 1564. [20] RAMIS, J.P ., Sous-ensembles analytiques d’une variété analytique complexe, Springer-Verlag (1970). [21] NOVERRAZ, Ph., Pseudo-Convexite, Convexite Polynomiale et Domaines d'Holomorphie en Dimension Infinie, North-Holland (1973). IAEA-SMR-11/25 MODERN THEORY OF BILLIARDS - AN INTRODUCTION G. GALLAVOTTI Istituto Matematico, Université di Roma, Rome, Italy Abstract MODERN THEORY ОГ BILLIARDS - AN INTRODUCTION. This paper presents an elementary introduction to the notions and ideas involved in the proof of the ergodicity of billiards. 1. BILLIARDS, DO YOU RECOGNIZE IT? Let N = [ 0, 1] X [0,1] mod 1 be a two-dimensional torus. Let Qj, Q2, . . . , Qm be m closed convex regions in N. Assume that Qj is a C2-smooth curve with never vanishing curvature and assume also that Qj П Q = ф, i f Í- Let V be the Riemannian manifold (with boundary) obtained by taking out of N the interior of Qj, . . . , Qm; the metric on V is the one inherited by N ( i.e . d s 2 = dx2 + dy2). Let TjV be the unitary tangent bundle of V. The elements of TjV can be thought of as applied vectors or as couples (q, 6), qeV and 0 s 0 S 2 n, where qGV is the point of application of the vector and в is its angle with a fixed direction. Define on TjV a probability measure p(dqd 0) = (const) ■ dq dS and a flow St, — oc 193 GALLAVOTTI qq' + qq = * x' = (q; 9 ) =s X = (q .6 ) FIG. 1. Construction of flow. : = (г,фГ = (г,'Ф'1 ( Û o\ V t‘ = Tx V^ ^ П$) о 00 FIG.2. Transformation T. IAEA-SMR-11/25 195 Let us define a transformation T: M -*• M as follows: choose xeM, think of it as the vector describing a billiard ball hitting 3Q and follow this ball back in time in the past until, at time т (x) < 0, it hits again Э Q in a point x' = Tx (Fig. 2). It is quite easy to check that the measure v (dr d 2. A RELATED PROBLEM In this section, we shall first discuss the idea behind the proof of the ergodicity of С-systems in a very simple case. This proof will be used to illustrate the necessity and the meaning of the various mathematical objects that have to be introduce^! to cope with the problem of billiards (as well as with the theory of the C-systems). 1 9 6 GALLAVOTTI The comparison system is going to be the much publicized — in many other papers of the Proceedings — authomorphism of the torus N = [ 0, 1 ] X [ 0, 1 ] mod 1 defined by т (x, y) = (x+y, x+2y) mod 1; see Fig. 5. Let e+, e_ be the two eigenvectors of the matrix (J \) anc^ let ^-+> = X+1 < 1 be the two eigenvalues. It is easy to check that the directions e+, e. are irrational, i.e. the tangent of the angle of these two directions with the x-axis are irrational numbers. The first claim is that the system (N, T, v dxdy) is a C-system and that the contracting and expanding foliations f c and f e consist of parallel straight lines (wrapped on the torus and parallel to e+ and e_). Let us note that the irrationality of the directions e+ and e. implies that each leaf of the foliation is dense in N. In fact, let (x, y)eN be a point and let Cçc (x, y) be the straight line parallel to e. and passing through (x, y). Let (x1, y')£C g (x, y); it is clear that rn (x, y) and t " (x\ y1) will be at a distance not exceeding their distance counted along the line t 11 Cê (x, y) which is d n (тп(х,у), т"(х\у')) = X+’nd ((x,y), (x',y')) t " c e (x, y) + cE (x,y) ~c ~c Similarly, one proves that the foliation Çe is contracting in the past ( i . e. under t ' 1) and that the line elements of f e locally expand in the past while the line elements of §e locally expand in the future. The expansion IAEA-SMR-11/2 5 197 and contraction coefficients are always X+or X_. This fact, of course, is responsible for the possibility of an easy treatment and visualition of the above system. We now show that (N, t , v) is ergodic using a proof which, though certainly not the simplest, is extremely instructive since it contains the main idea, due to Hopf, at the base of the proof of the ergodicity of C-systems and billiards. 3 . PROOF OF THE ERGODICITY OF ( j *). Let f be a continuous function feC(N). Then, by the Birkhoff theorem, the lim its n n i= 0 exist almost everywhere and are almost everywhere equal: f+(x) = f"(x) for xeU, v(U) = 1. We have to prove that for all feC(N) the functions f+(x) and f'(x) are constant almost everywhere. This, of course, implies the ergodicity of (N, t , v). Consider a covering of N with squares Ulf U2, .... with sides not exceeding 1 Д/2 and parallel to e+ and e_. We shall choose the squares in such a way that they overlap in chain (i.e. if P, QeN there is a chain U¡ , , v U•:i 2 . . ., UiLi such that (Ui DU )> 0 and Р е Ц , QeU,- ). The family {U¡} can be chosen to contain a finite number of elements. Clearly, it will be sufficient to prove that f+ and f" are constant on each Uj (almost everywhere). So let us fix feC(N) and show that its averages in the future and in the past are almost everywhere constant on U1; say. If PEU! let CEe(P) be the expanding leaf through P and let (P) be the connected part of C^ (P)DUi containing P; similarly, we define ü (P), se e F ig . 6 . e c In Ui, we use a system of orthogonal co-ordinates based on the vectors e+ and e_, which are parallel to the sides of Uj. If В is a measurable subset of Cf (P) or C{ (P), we denote by | В | its Lebesgue measure with respect to the abscissa or the line; hence, in particular, | C£ (P) | = | C£ (P) | = length of the side of Uj. с с Let us now consider the set ’e where U = (x/xeN, f+(x) and f' (x) exist, f+(x) = f"(x). 198 GALLAVOTTI FIG.7, P and Q with local contracting and expanding leaves. Since, by the Birkhoff theorem, v (U) = 1 (see above) it follows by Fubini's theorem that Ve has full measure i. e. y(Ve) =y(Uj). P ro o f: v (Uj) = v (UnUj), hence if s and s' are the co-ordinates of PeUj, P = (s, s'), we find v (Ux) = id s' Ids W P)=/ds,|V ° ’s,)nunui| which implies that for almost all s', we have |Cf (0, s'JOUflUil = |С^ (0, s ' ) and, again by Fubini's theorem, this implies that ] Cje(x)nUDUi) = 6 |c'ge(x)| for almost all хеиц in words, we can say that almost all points xEUj are such that the line Ce (x) lies almost entirely in UnUj. Similarly, we can define Vc and show that v (V ) = v (U ^. Let us now consider the set v = unu1nvenvc Clearly, V has full measure in Ujî v (V) = i/ÍUj). It is, therefore, enough for our purposes to show that f+(x) = f" (x) = const for xeV. Let P, QeV, then draw through P the local contracting leaf Cf (P) and through Q the local expanding leaf CÊg(Q) (see Fig. 7). ° The point T may or may not lie in V. In any case, it is possible to find a point P 'eC Êc(P) and a point Q 'eCÊe(Q) such that the points T', P1, Q1 are all in V (see Fig. 7). In fact, by construction, almost all points on the two lines Cêc(P) and C« (Q) lie in V (remember the choice of P and Q), hence as P' runs over VnC{c(P) and Q1 over VnCSe(Q)the point T' spans a set of full measure which, therefore, certainly intersects V. Now, the game is over; in fact, by construction: d (rnP , Tn P ' ) ------0 n-> <*• d (т~п P ', т 'пТ ') ------► 0 П-» Ж- d ( t 11 T 1, T n Q ’) ------► 0 d(T’ nQ ', t '"Q ) ----- » 0 n-> «. IAEA-SMR-11/25 199 (with exponential speed). Hence, by the uniform continuity of f, f( r n P) - f (rnP ') ------» 0 n -*■ f(T‘ nP ') - f ( r 'nQ') ----- »• 0 etc. n-> » Therefore, f+(P) = f+(P'); f'(P') = f(T '); f+(T') = f+(Q'); f'(Q') = f"(Q); but, by construction, it is also true that P, P1, Q, Q1, T 1 are in VCU hence f+(P) = f'(P); f+(P') = f'(P'); f+(T') = f'(T'); f+(Q') = f'(Q'); f+(Q) = f'(Q). Hence, all the above mentioned values of f+ and f" coincide; in particular, f +(P) = f+ (Q) = f"(P ) = f ‘ (Q) which means, by the arbitrarity of P and Q, that f+ and f are constant on V (and, therefore, almost everywhere). 4. HOW TO GENERALIZE THE ABOVE ARGUMENTS Obviously, in more general situations, things are not so nice and easy; nevertheless, the proof of ergodicity for the case of С-systems or even billiards system s proceeds along the same lines as the above proof of the ergodicity of (J gi lt is possible, in those cases, to construct a denumerable family of measurable sets {U j, forming a basis for the Borel sets, such that, given any two points x, y in a suitable set of measure 1 , one can find a finite number of sets Uit, U¡2, .. ., U¡ overlapping in chain (i.e. v (Uij П Uij + 1) > 0) such that x e U ii and yeU ¡m (see F ig . 8). Furthermore, to each of these sets U¡ the Hopf idea can be applied: in fact, roughly speaking, the sets U¡ can be thought of as unions of pieces of leaves of a contracting foliation and, at the same time, as unions of pieces of leaves of an expanding foliation; furthermore, onezcan use a "system of co-ordinates" based on the hypersurfaces C^nU ; = Cic and CEe П U¿ = Cie and the measure of a set BcU¡ can be computed as a double integral on the product of the natural m easures dg- a, dj- cr induced by the Riemannian metric on CE6 C or C .se . ic le More precisely, it is possible to construct two measurable partitions fc and Çe for each set Ui which are local contracting or local expanding 200 GALLAVOTTI leaves, and these two partitions are absolutely continuous with respect to each other (see below). The above discussion should be understood as an intuitive anticipation of the precise definitions to be introduced in the next section. 5. MEASURABLE FOLIATIONS Here we provide the precise definitions needed to fully understand the sentences of the last section. L et (M, t , v) be a dynamical system. Definition 1 : a measurable set UcM is said to be measurably par titioned by a k-dimensional foliation g if: 1 ) I is a partition of U. 2) The elements С ^ Щ are open k-dimensional piecewise smooth manifolds homeomorphic to an open k-sphere1. 3) If v denotes the restriction of v to the subalgebra Yj (? ) of the B o rel algebra in U consisting of the measurable sets that are unions of elements of § then v (B) = J v ( d C E) J p^(u)d? (j VB С U cU CgflB 1 6 where pg- (u) > 0 almost everywhere with respect to the natural measure (surface measure) d^ r on the manifold. f Remark: the above double integral has to be understood in the usual sense, i.e. p ~ (u) d ~ ( S E cf Пв must make sense and must be an integrable function with respect to v for all measurable BcU. Definition 2: If in U there are two measurable partitions f 1( g2, we say that g2 is absolutely continuous with respect to fj if every element of | 2 intersects every element of fi in just one point and if all We2H(?2) such that y(W) > 0 are such that Tg- (WnC{ > 0 for all apart for a set of C 's which are, however, contained in a null set. In this case, we say Ç 2<< ■ 1 It is perhaps important to state explicitly what is meant by an open piecewise smooth k-dimensional manifold (see Ref. [1] ). A closed k-dimensional smooth submanifold of a manifold M is a cf-smooth submanifold N homeo morphic to a closed k-sphere through a mapping ■which, in the neighbourhood of each point of N, is given in local co-ordinates by C2 functions having a limit on the boundary of N together with their derivatives; further more, if к > 2, the boundary 9N must consist of a finite number of closed smooth k-1 dimensional manifolds. A closed submanifold is called a closed piecewise smooth submanifold if it consists of the union of a finite number of closed smooth submanifolds and if it is homeomorphic to a closed k-sphere. An open piecewise smooth submanifold is a submanifold which is homeomorphic to an open к-sphere such that its closure is a closed piecewise smooth submanifold of the same dimensionality k. IAEA-SMR-11/25 201 Remark: if « Ç2 arid f 2 «_Çj , then it is quite clear that, apart for a "bad" set of local leaves Cfl and Cfz contained in a null set^the inter sections of any set W of full measure in U with Cfi£ and C£ e Ç2 have full measure with respect to the natural measures dg- cr and dg- cr, i.e. Fi к a? (WnC ) = erg- (C ) fi 1 Fi if the C£ are not "exceptional". si Definition 3: Let (M, v, t ) be a dynamical system and let f be a measurable decomposition of a measurable set UcM. Then f is said to be "contracting" if for any two points x, у chosen almost everywhere (with respect to the natural measure dg^cr) on a leaf C£ are such that d(Tnx, т пу ) ----- *-0 with the possible exception of a set of С 's contained in a null set. Similarly, we define an expanding measurable partition, i.e. in the above definition we replace т by r'1. From the above remarks and definitions we realize that (M, t , v) is going to be ergodic if it is possible to construct enough sets U admitting measurable foliations of contracting and dilating type which, furthermore, are absolutely continuous with respect to each other. 6 . HEURISTIC CONSTRUCTION OF THE FOLIATIONS FOR BILLIARDS Let us conclude by discussing how one can attack the problem of finding the contracting and dilating foliations in the case of billiards. We shall only present heuristic arguments. Suppose there is a contracting curve у through xeM. Then the mappings T"1, T"2, .... must all be smooth on у (remember that, by our conventions, T sends back into the past and T _1 sends into the future). From geometric arguments, it is easily shown that ifcp = cp(r) is the equation of a smooth curve Г and cp'= cp' ( r') is the equation of T Г, then d Hence, if T "1 is smooth on у for i = 1, 2, ... we find, by repeated application of the above formula, 4^(x) = -cos Rcos^T'M т(тЛ )+ ------1 I____ + Rcoscp(T”zx) - - kc(x) COSip(x) 202 GALLAVOTTI where кс (x) is the function defined by the continued fraction inside the parentheses; it converges since |т(Т"‘х)|ё ro>0 because Q¡nQj = ф, i f j. It is also not difficult to see from the above equation that the curve 7 (if existing) must be such that the distance between the T "1 images of any two points on 7 tends to zero as i -» cc (i.e. 7 is actually a contracting curve). The first real problem is to show that the above differential equation actually has a solution; this seems to be a difficult problem since it is easily realized that k c (x ) is discontinuous over a dense set and, on its set of continuity, it is not at all nicely behaved. However, it is possible to prove that if the distance of T"'x from the singularity points of T does not tend to zero too fast as i -* «, then the equation for 7 has a solution in a neighbourhood of x. A sim ilar construction provides the pieces of the expanding leaves. The set of the x in M for which T_1x does not get "too" close to the singularity lines of T can be shown to have measure 1, so that the differential equation defining the contracting leaves has a solution for a set of initial data having full measure [ 3 ] . At this point one has to construct a family {U j} of measurable sets which are measurably decomposable by expanding and contracting foliations. The constructions and proofs related to this point are a very nice piece of geometry and really at the heart of the theorem. They are rather similar to the analogous constructions encountered in the theory of the C-systems [2, 3] . However, let us note the basic conceptual difference between C-systems and billiards: for С-system s, each point x belongs to contracting and dilating leaves of foliation and, furthermore, the dependence of the leaf upon the point x is smooth. In the billiards case, we have a situation in which the leaves of both the contracting and dilating foliations cover M only almost everywhere; furthermore, they typically end on singularity lines of T or T "1 or iterates of them and, if y ( x ) is a leaf through x, its dependence on x is far from being smooth. The interested reader is referred to the famous papers [1-3] . REFERE NCES [1] SINAI, Ya, Dynamical systems with countable Lebesgue spectrum, II: Am. Math. Soc. Trans. 2 (1968) 34. [2] ANOSOV, D. , SINAI, Ya. , Some smooth ergodic systems, Russ. Math. Surv. : 22 (1967) 103. [3] SINAI, Ya. , Systems with elastic reflections, Russ. Math. Surv. 25 (1970) 137. IAEA-SMR-11/26 SOME REMARKS ON QUASI-ABELIAN MANIFOLDS F. GHERARDELLI, A. ANDREOTTI Istituto Matematico, Université di Firenze, Florence, Italy and Istituto Matematico, Universitá di Pisa, Pisa, Italy Abstract SOME REMARKS ON QUASI-ABELIAN MANIFOLDS. Some results obtained in the study of quotients of Cn by discrete sub-groups Г of non-maximal rank are presented. In this paper, we shall present some results obtained by the authors on studying quotients of (Cn by discrete sub-groups Г of non-maximal rank. The results we present are very simple and have been obtained by elementary m ethods. 1. Consider a complex Lie group G without non-constant holomorphic functions. Then it is well known that a) G is Abelian and of the form (En/ r n+m where Г is a discrete (Abelian) sub-group of Œn of rank n+m , n (Cn/rn+m=^rn+mx Rn-m where is a real torus of real dimension n+m in which F has a dense im age. The study of the quotients <Еп/Гп+т(0 < m < n) was initiated by Cousin ([2, 3]), who, in particular, studied the case n = 2, m = 1. Recently, the study of these quotients as non-compact Abelian Lie groups was again taken up by Morimoto in Refs [4, 5], where all the abovementioned properties are derived. 203 204 GHERARDELLI and ANDREOTTI 2. In the study of complex tori, it is a classical problem to ask for the conditions for a complex torus to be an algebraic manifold, i. e. an Abelian manifold. In analogy, we want conditions for X = Œn/ r n+m,0 < m < n, to be algebraic quasi-projective, i. e. a Zariski open set of some projective algebraic variety. Let 7r:(Cn-»X the natural projection; following the classical approach, in the case of complex tori ([1 , 9.1) we want conditions for the existence on of a non-degenerate meromorphic function f, such that 7r*f can be represented as a quotient of two entire functions having the properties of theta functions with respect to Г. Following the classical procedure step by step, we obtain the following necessary conditions for the existence of f: A) There exists a Hermitian form ^fon Œn X (Cn such that ^ fxt positive and B) ^¡rxr has integral values. Let С be the matrix of a basis of Г, then, in matrix form, condition B) can be written ‘С H С - *С ‘Н С = 2i A (1 ) where H is a matrix of the Hermitian form ,g^and A is a skew-symmetric square matrix of order n + m with integral entries. Remarks: 1. If m =n, conditions A) and B) are the classical Riemann relatio n s. 2. A) and B) are fullfilled if X = Œn/F n+mhas a Hodge metric, i. e. a Káhlerian metric such that the associated two-form has integral periods. Definition 1: If conditions A), B) hold, we say that X is a quasi-Abelian manifold. For a quasi-Abelian manifold we have the following Theorem 1: Let X be a quasi-Abelian manifold; than X is a covering of an Abelian manifold. Corollary: The existence of a Hodge metric on X is equivalent to conditions A), B) and also to the existence of a non-degenerate meromorphic function on X of the previous type. Sketch of the proof. The Hermitian form is determined by conditions A) and B) only up to a Hermitian form Asymmetric on Г X Г. It is then an exercise in linear algebra to show that there exists a vector 7 G(Cn such that i) The group 1} generated by Г and у h as a rank n + m + 1 over IR, ii) Fj ®IR contains a complex sub-space Tj of dimension m + 1 , iii) 9" can be chosen in such a way that ^ f + ^ ¡ r lxFl is positive and Im [gtf + S?1) has integral values on Tj X Гг. Repeating this argument n-m times we obtain the theorem. 3. The rank of the skew-symmetric integral matrix A in Eq. (1) is an even number not less than 2m: 2m S rank A S n+ m. Definition 2: The quasi-Abelian manifold X = (Dn/ r n+mis called of kind p if rank A = 2m + 2p (0 s2p sn -m ). IAEA-SMR-11/26 205 Then we have: Theorem 2: Let X = Œn/E n+m be a quasi-Abelian manifold of kind p. Then X is a fibre bundle over an Abelian manifold Y of dimension m + p with fibres (EPX ((C*)n"m"2P (0S2pSn-m ). In other words: Every quasi-Abelian manifold is an Abelian extension of the linear group (EPX( <ц*)п"ш_2Р by an Abelian manifold. This proves that our quasi-Abelian manifolds are essentially those considered under this name by Severi [ 8]. T h eir study as extension of algebraic groups can be found in Serre [7] where also other references are given. 4. Theorem 1 and 2 are obvious in the following example. Let X = (Е2/ Г 3; here conditions A) and B) are always fulfilled. By linear transformations in Œ2 the matrix С of Г can be reduced to the form w here a, ¡3 are two complex numbers so that X is without non-constant holomorphic functions if and only if the numbers 1 ,a,f¡ a re linearly independent over the rationale. The real space Г ® IR = R 3 contains a complex line F on which, at most, one generator of Г is lying. Suppose, for simplicity, that Г does not contain any generator of Г and also that Im a > 0. Then the matrix С (or the group Г) can be completed with another column vector ^^such that is symmetric and is a Riemann matrix, and this is the content of Theorem 1. Let now Uj, cj2, u3 be a basis of Г; project Œ2 over F in the direction of the complex line defined by щ. Then the projections of u2 and 103 define a lattice 7 on F. Passing to the quotient (Еп/Г we obtain a projection of X onto the elliptic curve F /7 with fibres Œ* (as stated in Theorem 2). 206 GHERARDELLI and ANDREOTTI Remark that the previous construction can be done in an infinite number of ways: we can start from any basis of Г. By direct computation it is easy to show that the elliptic curves which we obtain by these projections are, in general, non-isomorphic so that (С2/Г 3 is fibred over infintely many elliptic curves and we have a simple example of a (non-compact) complex manifold with infinitely many algebraic structures underlying the same complex structure (compare Ref. [ 6], note p. 34). REFERENCES [1] CONFORTO, F., Abelsche Funktionen und algebraische Geometrie, Springer, Berlin (1958). [2] COUSIN, P., Sur les fonctions périodiques, Ann. Sc. Ec. Norm. Sup., Paris 19 (1902). [3] COUSIN, P., Sur les fonctions périodiques de deux variables, Acta Mat. 33 (1910), [4] MORIMOTO, A ., Non compact complex Lie groups without non constant holomorphic functions, Conf. in Complex Analysis, Minneapolis (1966), [5] MORIMOTO, A., On classification of non compact complex abelian Lie groups. Trans. Amer. Mat. Soc. 123 (1966). [6] MUMFORD, D ,, Abelian varieties, Oxford (1970). [7] SERRE, J. P ., Groupes algébriques et corps de classes. Paris, Hermann (1958), [8] SEVERI, F ., Funzioni quasi-abeliane. Pont. Ac. Sc. (1947). [9] SIEGEL, C .L ., Analytic functions of several complex variables. I. A.S. (1948). [10] WEIL, A ., Introduction à l'étude des variétés kahlériennes. Paris, Hermann (1958). IAEA-SMR-11/27 LIFE AND DEATH OF THE BERNSTEIN PROBLEM E. GIUSTI Department of Mathematics, Stanford University, Stanford, C alif., United States of America Abstract LIFE AND DEATH OF THE BERNSTEIN PROBLEM. The Bernstein problem as well as some of its proofs and extensions are presented. 1. In 1915, Bernstein [2] proved his celebrated result concerning entire minimal graphs: Bernstein's Theorem. Let u(x,y) be a C2 function in IR2, solution to the minimal-surface equation: u ' = о (1) эх 4 ^ 2^ у эу in all of ]R2. Then the graph of u is a plane. Since the appearance of Bernstein's paper, several proofs of the theorem have been published; the following one, due to Nitsche [ lijáis one of the most elegant: Let us observe first, with Heinz, that Bernstein's result follows from a theorem, originally proved by Jorgens, stating that if Ф(х,у) is a function verifying the equation 2 Ф XX Фyy - Фxy = 1 (2)' > in IR2, then Ф must be a polynomial of the second degree. The link between this result and Bernstein's theorem lies in the fact that the minimal-surface equation, which can be written as (1 + u2)uxx - 2uxuyuxy + (1 +u2)uyy = 0 (3) is the necessary and sufficient condition for the existence of a function Ф(х,у) such that 1 + u‘ ■J 1 + u У фху= —UxUy .... . (4) Vl + u 2+ u 2 1 + uj b y - г-----¿----2 2 2 */l + u x + u у 207 208 GIUS TI Such a function Ф is easily seen to verify the hypothesis of Jôrgen's theorem, and hence to be a polynomial of the second degree. This implies, in turn, that the derivatives of u are constants. We come now to Nitsche’s proof of Jôrgen's theorem. We can suppose that Ф(х, y) is a convex function, so that the map J 5 = x + Фх(х,у) l Г) = у + Фу(х,у) is a diffeomorphism of IR2 onto itself. If we put Ç = I + Í17, and w (0 = x- Фх(х, y) - i(y- Фу(х,у)) (here x and у have to be understood as functions of f and rj), the function w(f ) is an entire holomorphic function. In addition, we have 1 - |w '(H |2 *2 + Ф XX + *Ф y y from which follows that w'(£) is bounded, and hence constant by the Liouville theorem. On the other hand, we have I 1 - W' I2 Ф - • 1 - |w- I2 = _[l± w ^ yy I 12 1 - I w' I so that Ф is a polynomial of degree two. 2. It is natural to look for an extension of the Bernstein theorem to dimensions higher than two, i.e. to ask whether or not an entire solution to the minimal-surface equation in H n U*i [ s ; ( 7 = s ) - " (I Du I denotes the length of the vector grad u) must necessarily be linear. On the other hand, most of the different proofs of Bernstein's theorem are based on complex function theory and hence cannot be extended to more than two dimensions. It was only in 1962 that Fleming [ 8 ] found a new proof of the theorem, using a method independent of the number of dimensions, and opening the way for further developments. 3. To sketch Fleming's idea, let us describe briefly De Giorgi's formalism for minimal surfaces. IAEA-SMR-11/27 209 Let A be a Borel set in ]Rn+1 and let ipA be its characteristic function. We say that the boundary of А, ЭА, is an oriented hypersurface (briefly: a surface) if the derivatives of ¡pA, in the sense of distributions, are Radon m easures. In this case, for every Borel set K, we define the area of the part of the surface ЭА lying in К as the total variation of the vector-valued m easu re D к The reason for the above definition will become apparent if one considers a set A with a smooth boundary. Let g(x) be a smooth vector-valued function with compact support; we obtain.from the Gauss-Green formula: A За w here v is the outer normal vector to ЭА and dHn is the surface measure. On the other hand, we have /div g dx = / / D and hence (8 ) so that, for regular sets, our definition coincides with the usual surface m e a s u r e . The following definition of minimal surfaces is now quite natural: Definition. Let fi be an open set in ]Rn + 1. We say that ЭА is a surface of least area, with respect to fi, if for every set B, which coincides with A except possibly in some compact set Kcfi, we have D If ЗА is a surface of least area, we shall call A a minimal set. Let us remark that the reason for introducing the compact set К in the definition is that a surface of least area may have infinite area (like, e.g. a hyper plane) . The relation between minimal sets and solutions to the minimal surface equation lies in the fact that a C2 function u(x) in IRn is a solution to Eq. (6 ) if and only if the set (9) has the boundary of least area in IRn+1. 210 GIUSTI 4. The idea of Fleming is the following: Let t > 0, let A be the set (9) and let At = | zeH n+1 :tzeA It is clear that 9At is a surface of least area in IRn+1, for every t > 0. We can choose a sequence {tn}, tn oo, such that the sequence <рд converges in L loc (K n+1) to the characteristic function tpc of some set C. n The set С is a cone and, what is crucial, if ЭС is non-singular (i.e. if ЭС is a hyperplane), then ЭА itself is a hyperplane. In this way, the extension of Bernstein's theorem to IRn is reduced to the problem of the existence of singular minimal cones in IRn+1. Since no such cone can exist in ]R3, Fleming's argument gives a new proof of the Bernstein theorem. 5. The next step was taken by De Giorgi [ 7] . He showed that the limit cone С cannot be singular only at the vertex. In fact, if С is singular, then it must be a vertical cylinder: С = С' X R over a minimal cone С' С K n. This extends Bernstein's theorem to graphs in R 3, since now the non existence of minimal cones in IRn implies the Bernstein theorem for minimal graphs over IRn. 6 . The importance of minimal cones cannot be fully understood without mentioning the problem of the regularity of surfaces of least area. Let T2be an open set in IRn and let A be a minimal set in Г2. We can suppose that 0ЕЭАПГ2. Let us consider, as before, the sets At = x e E n : txeA We can now let t -*■ 0 through a sequence tn in such a way that 7. In 1966, Almgren [1] proved the non-existence of singular minimal cones in IR4, and in 1968 Simons [12] extended this result up to IR7. The idea of Simons consists in evaluating the eigenvalues of the bilinear form which gives the second variation of the area for a stationary cone. A careful estimate shows that, for n s 7, the first eigenvalue of this form is negative, and hence proves the existence of variations which decrease the area. In the same paper Simons gives the example of the cone Cr< -jxelR - f td 8 :x 2,1+x 2,2+x 2,3+x 24 <. x 2, |.+ x 2,¡1 + x, 2, + x8j2] which is locally stable, in the sense that every compact variation of С initially increases the area. This cone is actually a minimal cone (in the sense of our definition), as shown by Bombieri, De Giorgi and Giusti [3], and provides a counter example to the problem of interior regularity of minimal surfaces. With the help of the cone С and of some calculations, the same authors were able to construct complete minimal graphs over IRn, for n è 8, different from hyperplanes. IAEA-SMR-11/27 211 8 . We can summarize the previous discussion as follows: Theorem A. Let u(x) be a C2 function in IRn , solution to the minimal-surface equation } Then, if n S 7, the graph of u is a hyperplane. This is not the case for n § 8. 9. The preceding theorem gives a complete solution to the Bernstein problem in IRn, but since the answer is negative for higher dimensions, naturally, some additional questions arise. A first kind of problem is the search for additional conditions on u(x) that imply, of course, for a number of dimensions higher than seven, that the graph of u is a plane. We state here two results in this direction: a) Moser [ 10]. If, in addition to the assumptions of Theorem A, we suppose that the derivatives of u are bounded, then u is linear. b) Bombieri, De Giorgi and Miranda [4]. If the function u(x) satisfies the inequality u(x) È - к (1 + I x I ) then u is linear. The proof of result b) depends on the a-priori estimate: (10) for the gradient of positive solutions to the minimal-surface equation in the ball of radius R. We remark that the estimate (10) guarantees the regularity of generalized non-parametric minimal surfaces (see, e.g. Ref. [9]). 10. Another type of question concerns the behaviour of complete minimal graphs at infinity. Very little is known on this subject; we refer here to a paper of Bombieri and Giusti [ 5] . Let S be the surface graph of u, let OeS and let Sr = SnBr be the part of S lying in the ball of radius r. Iî0is the Laplace operator on S, we have the following result: Let g(x) be a positive superharmonic function on S (i.e. ® g s 0). Then i |2 -1/2 In particular, since i/(x) = (1+ |Du| ) is superharmonic, we easily obtain 1 + I Du I dx (И) 212 GIUS TI w here я, is the projection of Sr on IRn: |{ x e E n: Ix 12 + |u(x) |2 < r 2} From (11) one can derive the estimate ln (12) In fact, let p > 0 and let r 2 = p2 + sup | u(x) |2. I x | < p The ball of radius p in K n is contained in 7rr, and hence v m eas r È u n pn On the other hand, n и r n + 1 and inequality (12 ) follows at once. Inequality (12) is, in some sense, an improvement of inequality (10); let us point out explicitly that it is only valid for complete minimal graphs. We remark that the inequality sup I x | < p (with exponent 1 on the right-hand side) would imply that |u(x)| Sk(l+ |x|) 4 and hence a growth condition on complete minimal graphs. Although n does not seem to be the best exponent and some improvement can possibly be obtained, the way from inequality (12) to inequality (13) seem s to be quite long. REFERENCES [1] ALMGREN, F. J. , Jr., Some interior regularity theorems for minimal surfaces and an extension of Bernstein's theorem, Ann. Math. _85 (1966) 277. [2] BERNSTEIN, N.S. , Sur un théorème de géométrie et ses applications aux équations aux dérivés partielles du type elliptique, COmm. Soc. Math, de Kharkov (2ème sér. ) (1915-1917), 38. (See Math. Zeit. 26 (1927) 551, for a German translation.) [3] BOMBIERI, E. , DE GIORGI, E. , GIUSTI, E. , Minimal cones and the Bernstein problem, Inv. Math. 1 _ (1969) 243. [4] BOMBIERI, E. , DE GIORGI, E ., MIRANDA, M. , Una maggiorazione a priori relativa aile ipersuperfici minimali non parametriche. Arch. Rat. Mech. Anal. ^2 (1969) 255. [5] BOMBIERI, E ., GIUSTIv E ., A Harnack's type inequality for elliptic equations on minimal surfaces, Invent. Math. JL^(1972) 24. [6] DE GIORGI, E. , Frontiere orientate di misura minima, Seminario Matemático Scuola Norm, Sup. Pisa (1960-1961). Editrice Técnico Scientifica Pisa (1961). IAEA-SMR-11/27 213 [7] DE GIORGI, E. , Una estensione del teorema di Bernstein, Ann. Scuola Norm. Sup. Pisa (3) 19 (1965) 79. [8] FLEMING, W .H., On the oriented Plateau problem, Rend. Circ. Mat. Palermo (2) 11 (1962) 69. [9] GIUSTI, E. , Superfici cartesiane di area minima, Rend. Sem. Mat. Fisico Milano 40 (1970) 1. [10] MOSER, J. , On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961) 577. [11] NITSCHE, J.C .C ., Elementary proof of Bernstein's theorem on minimal surfaces, Ann. Math. 66_ (1957) 543. [12] SIMONS, J. , Minimal varieties in Riemannian manifolds, Ann. Math. (2) 88 (1968) 62. IAEA -SMR-11/28 INVARIANTS OF FOLIATIONS C. GODBILLON Département de mathématique, Université de Strasbourg, France Abstract INVARIANTS OF FOLIATIONS. The definition of a foliation is given. An invariant of foliations of codimension one is introduced and relations with the Gelfand-Fuks cohomology are established. INVARIANTS OF FOLIATIONS We give here some elementary ideas on the new cohomological invariants of foliations introduced in Ref. [4] . For further developments, se e R efs [1, 2 ]. 1. Foliations [5] Consider first a differentiable manifold M of dimension m, and a smooth vector field X on M without singularities. Each maximal integral curve of X is then a connected one-dimensional submanifold of M, and the fam ily (0 £) of these orbits has the following properties: i) M = U 0{; ii) Oc П oj = ф fo r t j; iii) if EXCTX(M) is the tangent space at x to the orbit throughx(Ex is the subspace of TX(M) generated by X(x)), then E= и Ex is a smooth subbundle of rank one of T(M). xeM Such a situation is a foliation of dimension one of M. More generally: Definition. A foliation & of dimension q (or of codimension m-q) of M is a family (F{ ) of connected submanifolds of dimension q of M having the following properties: i) M = U{ F{ ; ii) F{ П Fj = ф for ¿ ф j; iii) if Ex с TX(M) is the tangent space at x to the submanifold F{ containing x, then E = и Ex is a smooth subbundle of rank q of T(M ). xeM E ach Fg is a le af of ,9^ and E is the tangent subbundle of ¡P~. E x am p les. If M = Q X N, the fam ily (Q X {n l ) neN is (if Q is connected) a foliation of M. More generally, if p : M ->• N is a submersion of Mm into Nn, m s n, the family of the connected components of the inverse images p'Mx), xeN, is a foliation of codimension n of M. 215 216 GODBILLON If & is a foliation of codimension one of M its tangent subbundle E can be (locally) defined by a Pfaffian form (i. e. a differentiable form of degree one) со without singularities: Ex= {e G TX(M)| Frobenius theorem. If uAdu = 0, there exists for each point x e M an integral submanifold of и containing x. Moreover, if Nj and N2 are two such integral submanifolds, there exists an integral submanifold P of u such that (in some sense) PDNj U N2. This allows us to introduce the notion of "maximal" integral submanifold of u; and the family of these submanifolds is a foliation of codimension one of M whose tangent subbundle is (globally) defined by u. One says that u is an integrable Pfaffian form. Examples: i) If u is a closed Pfaffian form without singularities (du = 0) on M, u is integrable. Moreover, if M is compact all the leaves of the corresponding foliation are diffeomorphic [5]. ii) Let N be a manifold with boundary, and f be a differentiable function on N having 0 as a regular value and such that f_1(0) = 3N. Then u = df+ fd0 defines a foliation of codimension one of M = N X S 1 having the boundary 3M = 3N X S 1 as a union of leaves. iii) Taking N = D2 (the two-dimensional disk) in the preceding example one gets a foliation of codimension one of the torus D2 X S 1. Then, gluing two such foliated toruses by identifying the meridians of the boundary of one with the parallels of the boundary of the other, one obtains a foliation of codimension one of the three sphere S3: the Reeb foliation of S3. 2. An invariant of foliations of codimension one [4] Let S^be a foliation of codimension one of M defined by a Pfaffian form u, Since uAdu = 0 one can write du = uAuj. By differentiating one gets 0 = uAduj ; so that duj is, too, divisible by u: duj = uAu2. Consider then the three-form Г2 = -UjAdUj = UAU|AU2. It is closed (dfi= 0); and thus it defines a cohomology class [ Q] in the de Rham cohomology group H3(M,R). Proposition 1. The cohomology class [Г2] depends only on Proof i) If du = uiluj one has uA(ux - иг') = 0; and, therefore, u •[ =uj+fu dUj' = dUj + df Au + fdu - Uj'AdUj1 = - UjA duj - u-l A df A u = Q - d(f du). ii) If u" is another Pfaffian form defining , one has u" = gu, where g is a differentiable function on M without zero. Then du" =u"Auj' where u " = Uj - dg/g, and dir i i - Uj'AdUj = -UjA dUj + A dw^ = П + d(Log | g | dUj ) q. e. d. IAEA-SMR-11/28 217 Examples, i) If и is closed [Í2] = 0. ii) If u = df + fdS, as in example ii) of section 1, u = dô and [Г2] = 0. a b iii) Let G = SL(2, R) = ad - be = 1 f. {(:.с d.;) }■ There is a basis a , о?}, or2 of the space of left invariant Pfaffian forms on G such that da = of A dffj = a A a 2 d a 2 = ° 1Л а 2 The form a is thus integrable and defines the foliation <3 whose leaves are the left cosets of the subgroup and A = a A a 1A a 2 is a volume form on G. There exists a closed discrete subgroup Г of G such that the quotient manifold M= F\G is compact. The foliation Ç& of G being left invariant induces a foliation 5^of M for which Í2 is a volume form; and, therefore, [Г2] =j= 0 (this important example is due to Roussarie). iv) Thurston has shown that on S3 the invariant [f2] can be any cohomo' c la s s in H3(S3, R) [6]. 3. Relations with the cohomology of Gelfand-Fuks [3]. Differentiating the relation du1= шЛи2 one obtains 0= uA^AUg - du2), so that du2 = U jA u 2 + u A u 3, It is possible to iterate this procedure and introduce Pfaffian forms u)k , k G N. These forms are not uniquely determined, but one has 1. J where the coefficients c^ . are "universal": i, J for i+j ^ k+1 for i+j = к + 1 218 GODBILLON L e t л / Ъ е the real Lie algebra with basis {Y0, Y1(. . . , Yk, . . and Lie bracket [Y. , Y. ] = V ck Y = i (l + j ~ 1)'~ Y 1 ■> L i.J к 2 it -, i > ++ i-J l b * J * If X k = 2k; Yk one has [X¿,Xj] = ( j- i) X i+j_1 ; so that is isomorphic to the Lie algebra of formal vector fields in one variable: jrf = { x = V a xk -^-T, У a. x 1 , У b. xJ = У (j-i) a.b. x*+) 1 -Д- l z_, к 9xJ> _Z_, 1 Эх L—i J 9x_ z_, 1 J 8x Letj^'be the space of continuous linear forms on (i. e. forms which depend of a finite number of coefficients ak): has as a basis the forms :Iaixi4 ( - i r k '. a„ Denote by C * (jzt) = EC (j<0 the exterior algebra of (space of continuous cochains on já), and by d the usual coboundary operator in C*(jrf) given by One has d°d = 0; and the quotient space H* (,c/) = Kerd/Imd is the con tinuous cohomology space of the Lie algebra ji (Gelfand- Fuks cohomology of jd). The Lie algebra operates on Qr{j>i) by the Lie derivation Lx = ix d +diJ{ (where ix is the interior product); and one has v (l k >4 where K=4 d a = ) ck a A a к 1 _л i. ) ‘ J i, i So: Proposition 2. There exists a morphism of complexes h from C*(,rf) to the de Rham complex Л(М) of M such that h(aQ ) = u. This induces a morphism h* from the Gelfand-Fuks cohomology H*(jd) to the de Rham cohomology H*(M, R); and the class M is in the image of h*. In fact: Proposition 3 [3]. One has Hk(jj/) = 0 for к =^0,3 H3U ) = R with generator the class ofa 0Ao1 A^2. IAEA-SMR-11/28 219 Proof. Let H and К denote the formal vector fields x 9/Эх and Э/Эх. One has [H,KJ = -K. We say that a cochain a e C*(j^) is of weight p if LHa = pa. Then i) ak is of weight 1-k (LHL K = -L K +LRLH); ii) if a is of weight p and J3 is of weight q, then a A/3 is of weight p + q; iii) if a is of weight p, so are da (LHd = dLH ) and iHa (iHLH = L H i ). T h erefore C*(,jd) is the direct sum of the homogeneous subcomplexes С Cs/), where C p(jd) is the subspace of cochains of weight p. However, for p ^0 every cocycle in С (j¡/) is a coboundary: if LHa = pa and da = 0, then pa = diHa, so that the inclusion C 0(jrf)e-* C *^ ) gives rise to an isomorphism on the cohomology groups. The subcomplex C Q(j¡/) is zero in degree larger than 3, and is generated by 1 in degree 0 «j in degree 1 ajAoj in degree 2 agAajAttg in degree 3 with d ( « x ) = a 0 A a 2 d(l ) = d(aQAa2) = d(aQAal Aa2) = 0 so that Н кЫ ) = 0 for к =^0,3, H V ) = R, = R with the class of a ^ A a ^ A a ^ as a generator. This shows (proposition 1) that the morphism h* : H*(j ¡0 -► H*(M, R) depends only on ^ REFERENCES [1] BERNSTEIN, I., ROSENFELD, B.. Functional Analysis (1972). [2] BOTT, R.. HAEFLIGER, A., to appear in Bull.Am.Math.Soc. [3] GELFAND. I.. FUKS, D., Cohomology of the Lie algebra of formal vector fields. Izv. Akad.Nauk SSSR, Ser.Mat. 34 (1970) 322. [4] GODBILLON, C., VEY, J., Un invariant des feuilletages de codimension 1. С .R.Acad.Sci.Paris 273 (1971) A92. [5] REEB, G ., Sur certaines propriétés topologiques des variétés feuilletées, Hermann (1952). [6] THURSTON, W., Noncobordant foliations of S3, Bull. Am.Math.Soc. 78 (1972) 511. / IAEA-SMR-11/29 ON THE LOCAL SOLVABILITY OF LINEAR PARTIAL DIFFERENTIAL EQUATIONS H. GOLDSCHMIDT Université scientifique et médicale de Grenoble, Grenoble, France Abstract ON THE LOCAL SOLVABILITY OF LINEAR PARTIAL DIFFERENTIAL EQUATIONS. A survey of localsolvability results for linear inhomogeneous partial differential equations with С -coefficients, as well as conditions for solvability and non-solvability of such equations, are given, If x,.1 * ....* x n are the co-ordinates on Hn,’ we set and if a = (a^, . .. , an) is an n-tuple of non-negative integers, we set П and A differential operator of order m on an open set i2cE n is a linear operator where u is a complex-valued function or distribution and the a0, are complex valued C~-functions on S7, and some coefficient a0, with |ar| = m does not vanish identically on Г2. We shall consider the equation Pu = f (1) where f is a given complex-valued C“"-function on Q. Theorem 1 (Cauchy-Kowalewski). If the coefficients a0 of the operator P of order m and f are analytic in a neighbourhood of xQefi and if the coefficient of D™ is f 0 at x0, there exists an analytic solution u of Eq.(l) on a neighbourhood of x0. Theorem 2 (Ehrenpreis - Malgrange). If the coefficients a^ of the operator P on ]Rn are constant, then,for all fGC^ (Rn), there exists a C“'- solution u on lRn ofE q.(l). 221 222 GOLDSCHMIDT The situation for C“ -coefficients is quite different as is shown by the following striking example of Hans Lewy (195 7) on IE3: Pu = -iDjU + D 2u - 2(x1+ ix 2)D3u where i = . If f is an analytic function on a neighbourhood of x 0eIR 3, according to Theorem 1, there is an analytic solution of Eq. (1) on a neighbourhood of x0. If f is a function of the variable x 3 alone, then Lewy proved by elementary methods that, if f is of the form Pu on an open set U eIR 3 where u is a C“-function, f must be necessarily analytic on U. Choosing a С ““-function f of the variable x 3 which is nowhere analytic, an operator P and a function f are obtained for which Eq.(l) has no C^-solution and even no distribution solution u on any non-empty open subset of IR3. With such a choice of f, the homogeneous equation Pu - fu = 0 (2) has no non-trivial solution ueC 1 (fi) on any non-empty open subset fi of IR 3 . If u ^ 0 on fi, then v = log u is C 1 on some open set and satisfies Pv = f, leading to a contradiction. If f were analytic, there would exist non-trivial local analytic solutions u of Eq.(2) according to the Cauchy-Kowalewski existence theory. Recently, Nirenberg has given an example of a first-order operator of the form t 9 . , ,. 3 L = at +1 Pm(x, I) = £ aa(x)?° |n| =m w here x e fi, f = ( f j ...... Ç j e E " and ? “ = f “ i . . . . if a = K , . . . , <*„). We may consider Pm to be a complex-valued function on the cotangent bundle T* (fi) = fi X IRn of fi whose co-ordinates are (x, f)efiX IRn. Recall that if cp, ф are complex-valued functions onT*(fi), we may define the Poisson brack et , Л = V ( È Hormander has given the following necessary condition for the existence of solutions: Theorem 3. Suppose that Eq.(l) has a distribution solution u£® ' (П) for every fCC” (Г2). Then {P m< Pm[ (x - 5) = 0 whenever Pm (x, f) = 0, (x, ?)eT * (Q) (3) In the case of Lewy's example If fj = -2x2, S2 = ^xi ant^ 5з = 1. then Pj (x, f ) = 0 and {Pb P j} (x, f ) / 0; hence condition (3) does not hold for any non-empty open subset £2 of IR3. We now give the precise notion of local solvability. Definition. An operator P is said to be locally solvable in if for all x 0efi, there is an open neighbourhood UC£2of xq such that for any feC " (U) there is a distribution u£® ' (U) solution of Eq.(l). We shall consider only operators of principal type. Definition. The operator P of order m is of principal type in Q if, for аП (x, f)eT*(Q ), Ç f 0, one has dEPm(x, Ç) f 0. By Euler's identity, for homogeneous polynomials m Pm (x, f ) = Ç • gradE Pm (x, ?) and any zero of d^Pm will be a zero of Pm of multiplicity, at least, two. In particular, all elliptic operators, i.e. those operators for which Pmvanishes only if Ç = 0, and all hyperbolic operators are of principal type; the para bolic ones are not. Let us consider the simple example of an operator of principal type of the form (4 ) where (x, t) are the co-ordinates on IR2 and b(t) is a real-valued C°°-function on some open interval -T b(t) does not change sign in the interval |t| < T (5) If b(t) = tk, where к is a positive integer, then P is locally solvable in Q if and only if к is even. We wish to describe the condition of Nirenberg-Trêves which extends condition (5) to a general operator P of principal type. We write Pm (x, f ) = A (x, f ) + iB (x, f ) where A, В are real-valued functions on T* (Г2). Let n ЭА а ЗА 3 3?: ЭХ: ЭХ: 3?, ) j= 1 224 GOLDSCHMIDT be the Hamiltonian vector field of A on T* (Q). The integral curves of HA are the solutions of the equations • _ Э_А • _ Э_А X ' Э?j ’ ' ‘ 3xj and are called bicharacteristic strips. Along such a curve, A is constant. If the value of A on a strip is zero, then it is called a null bicharacteristic strip of A. The function HAB is just the Poisson bracket {B, A}, so that Hormander' condition (3) can be reformulated as (HAB) (x, Ç) = 0 whenever Pm(x, f) = 0, (x, f)e T * (í¿) o r a s: If В vanishes at a point (x, Ç) of a null bicharacteristic strip В does not change sign on any null bicharacteristic strip of A. We now state the Nirenberg-Trêves condition generalizing the previous condition: (N-T) For all (x0, f 0)GT* (Q), Ç0 f 0, there is a neighbourhood U of (xQ, Ç0) in T * (Г2) and a z£ С satisfying the following: (i) d|Re (zPm) (x, Ç) / 0 for all (x, Ç)e U; (ii) the restriction of Im (zPm) to any null bicharacteristic strip of Re (zPm) in U does not change sign. In verifying this condition for operators of principal type it is enough to take z = 1 or z = i. If P is of principal type and Pm has real coefficients, then this condition holds always. The following result was obtained by Nirenberg-Trêves when the coefficients of Pm are analytic in Q and by Beals-Fefferman in the general c ase : Theorem 4. Let P be a differential operator of principal type of order m in f2. If condition (N-T) holds, then P is locally solvable in Q. Nirenberg-Trêves also showed that if Pm has analytic coefficients, then condition (N-T) is also necessary: Theorem 5 (Nirenberg-Trêves). Let P be a differential operator of principal type of order m in Q. If Pm has analytic coefficients, condition (N-T) holds if and only if P is locally solvable in Г2. REFERENCES [1] HORMANDER, L. , Linear Partial Differential Operators, Springer-Verlag, Berlin, Gottingen, Heidelberg (1963). [2] TREVES, F. , On the existence and regularity of solutions of linear partial differential equations, Proc. Symp. Pure Math. , Amer. Math. Soc., Providence, R.I. (to appear). IAEA-SMR-11/30 COMPACT OPERATORS AND THE MINIMAX PRINCIPLE R. A. GOLDSTEIN, R. SAEKS Department of Mathematics, University of Notre Dame, Notre Dame, Ind., United States of America Abstract COMPACT OPERATORS AND THE MINIMAX PRINCIPLE. In the present work, the rudiments of a structure theory for a class of compact operators on a Banach space, based on a non-spectral minimax principle, are presented. 1. INTRODUCTION Since its inception, the pre-eminent problem of operator theory has been the formulation of a structure theory for operators on an infinite - dimensional space which extends, in a natural way, the classical structure theory for matrices. In the case of a compact operator on a Hilbert space, such a generalization is obtained by approximating it with a sequence of operators of finite rank (i. e. finite-dimensional range), thereby allowing one to deduce the structural properties of a compact operator on a Hilbert space from those of its finite rank approximates. Unfortunately, the lack of a suitable minimax principle for the spectrum of a compact operator on a Banach space, and hence an approximation theory, has precluded the use of such an approach in this more general setting. In the present work, the rudiments of a structure theory for a class of compact operators on a Banach space, based on a non-spectral theoretic minimiax principle are presented. Here, the minimax principle is used to define a sequence of operator widths, which after appropriate normalization are shown to play a role in the Banach space theory sim ilar to that of the eigenvalues and s-numbers of the Hilbert space theory. In particular, an approximation theorem and a trace class concept for compact operators is obtained which naturally extends the classical Hilbert space theory to Banach space. In the following section, the structure theory for compact operators on a Hilbert space is reviewed, first for positive hermitian operators and then for general compact operators. These results are then employed in the following section to motivate the concept of an operator width which after normalization is used to formulate an operator approximation theorem and to characterize various ideals of trace class operators. Finally, a number of open problems are discussed. 2. EIGENVALUES IN HILBERT SPACE For an operator, A, on a Hilbert space, we define its eigenvalues, in the usual manner, as the set of complex numbers, Xs. a(A), for which the operator [A-A.] fails to have a bounded inverse. In general, the complexity 225 226 GOLDSTEIN and SAEKS of the operator A manifests itself in the complexity of the set o(A). In particular, i) If A is bounded o(A) is bounded. ii) If A is compact c(A) is countable with zero as its only accumulation point. iii) If A is self-adjoint ct(A) is real. iv) If A is positive cr(A) is positive. As such, the eigenvalues of a positive self-adjoint compact operator form a non-increasing sequence [2] of positive real numbers, A;(A) = 0, 1, 2, ... , with the largest eigenvalue ( = ||a||) taken as \(A), the next largest as Xj(A), etc? Once the eigenvalues of A have been so ordered, their alternative characterization via the minimax principle [2, 3] is given by X¡(A) = minimum maximum J|Аф - l|| ( 1) LeLj IM -1 where L¡ is the set of i-dimensional subspaces and ||Аф-ь|| denotes the distance of the vector Аф from the subspace L. Now, with the aid of the above minimax theorem, the structure of A follows [2, 3]. In particular, i) A has the approximation property (i. e. A can be approximated in the uniform operator topology by a sequence of finite-rank operators). ii) A is a p-trace class operator if and only if X¡(A) is an i p sequence. iii) A has a diagonal representation Ax = У Х.(А)<^,х>ф. (2) i = 0 where фх is the i-th eigenvector of A. Unlike the positive self-adjoint operator considered above, a general compact operator on a Hilbert space need not have a positive real spectrum, hence the ordering of eigenvalues which makes the preceding development possible is not applicable in the general case. Fortunately, however, a structure theory for general compact operators can be deduced from the eigenvalues of its "magnitude". That is, one takes a polar decomposition [1] of a given compact operator, A, as A = VM (3) where M = -J A*A is positive and hermitian, and V is a partial isometry, and uses the eigenvalues of M to characterize A. Here, we denote by s¡ (A) the i-th eigenvalue of M, i. e. s¡ (A) = X¡(M) =Xi(Æ*Â) (4) 1 An eigenvalue of multiplicity m is repeated m times in the sequence; hence, the sequence is non increasing but may not be decreasing [2]. IAEA-SMR-11/30 2 2 7 and term the positive non-increasing sequence s¡(A), i = 0, 1, 2, ... , the s-numbers for A [2]. As before, the basic tool which allows the s-numbers to yield a structure theory is a minimax principle [2,3] yielding s¡(A) = minimum maximum ||Аф-ь|| (5) L<=L; М - 1 and, in turn, a structure theory for the general compact operator A. In particular: i) A has the approximation property. ii) A is a p-trace class operator if and only if s¡(A) is an i p sequence, iii) A has a "Schmidt" representation Ax = У Si(A)<<^i,x>0¡ ( 6) i= о where Ф; is the i-th eigenvector of M =\/A*A and =V ф1. 3. WIDTHS AND s-NUMBERS IN BANACH SPACE For compact operators on a Banach space the lack of an involution precludes the possibility of using a polar decomposition to reduce an arbitrarily given compact operator to one with positive eigenvalues as was done in Hilbert space. Fortunately, however, the representation of the s-numbers implied by the minimax principle of Eq. (5) does not employ the involution and is perfectly well-defined for operators on a Banach space. Therefore, rather than attempting to define the s-numbers for a compact operator in Banach space from its eigenvalues we use the minimax principle to define the s-numbers, though with an appropriate normalization [5]. For this purpose, we define the i-th width of a compact operator, A, on a Banach space, X, by dj(A) = infimum maximum || Аф - b|| (7) L e4 M - i and the i-th s-number by s¡(A) =Pi(X)d¡(A) (8) where Pi(X) = supremum infimum ||e || (9) L e L ¡ E g El where the infimum is taken over all projections, E, on L. Here, dj (A) is just the width of the image, under A, of the unit ball of X, in the sense of Kolmogorov [4] and the normalizing factor, p¡(X), is determined entirely by the Banach space independently of A. For a Hilbert 228 GOLDSTEIN and SAEKS sp ac e Pi(X) = 1 for all i, hence our Banach space definition for the s- numbers coincides with the classical definition of Eq. (4) [1]. In fact, Pi(X) = 1 if and only if X is a Hilbert space [6], hence the widths and s- numbers are indeed different in Banach space. In general, the p¡ (X) satisfy, [5,7,8], 1 S p . (X) Si (10) and may become unbounded. In fact, they grow linearly with i for Lp and lp, P ^ 2 [8]. Some elementary properties of the widths and s-numbers for compact operators on a Banach space are as follows [4, 5]: i) do(A) = s 0(A) = II a||. ii) di(A) is a non-increasing sequence of positive numbers which con verges to zero if and only if A is compact. iii) d¡(A) = S j( A ) for all i if and only if X is a Hilbert space. iv) pi(X) = Pi (X*) if X is reflexive. Since in a Banach space the widths and s-numbers differ there are two alternative formulations of the trace class operators which generalize the Hilbert-space concept [2, 5]. Hence, we say that a compact operator A on a Banach space X is of class Dp if d¡(A) is an t sequence and we say it is of class Sp if Sj (A) is an sequence, 1 S p S ®. Similarly, we denote by D„ the set of operators for which d¡(A) is a Co sequence and we denote by S^ the set of operators for which S¡(A) is a C 0 sequence. Clearly, Sp С Sq П П ISpSqSoo (11) Dpc D q and by analogy with the Hilbert space case [l] we may term an operator in class D: d-nuclear while clearly the operators in D„ are just the compact operators. Similarly, the operators of class Sj may be termed s-nuclear and those of class ^ may be termed s-compact. 4. AN APPROXIMATION THEORY An operator, A, on a Banach space, X, is said to have the approximation property if it can be approximated in the uniform operator topology by a sequence of operators of finite rank, and the Banach space, X, is said to have the approximation property if every compact operator on X has the approximation property. Although, (as once conjectured by Banach), it is not true that every compact operator has the approximation property [9], it is possible to show that the s-compact operators do indeed have the approximation property [5]. Theorem: Every s-compact operator has the approximation property. Proof: Let L¡ be a subspace of X such that + dj (А) й m axim um ||Аф - l || ( 12) GOLDSTEIN and SAEKS 229 Let L| be its annihilai or in X *, and let E¡ be a projection onto L; satisfy in g |м .{ГД ити } + 1 1131 i Now E^ = 1 -E r is a projection with range in ii. for if f = E^g =(1 -Er)g and x is in Lj then since E¡ is a projection onto L¡. Moreover, IK II = 111 -E?IIS 1 +||E?II = 1 + IIE¡ IIS 2 IIeJI S 2 (jfe™ um ||f||+ l } *2 f a (X) = l } (15) where the last inequality results from the fact that the supremum in the definition of p¡(X) is taken over all i-dimensional subspaces of X including L¡. Now upon recongizing [10, 11] that the distance between the vector А ф and the subspace Ц is given by IN - Lj I = maximum | and that the minimum in Eq, (12) is almost achieved by L if we have + di ^ maximum II Аф - l J = maximum maximum | = maximum maximum | Ilf IN I ||f II s i Here, the first equality is that of Eq. (12) with the minimum achieved for the specified Lj, the second results from substitution of the distance formula of Eq. (15), the third results from interchanging the order of the maxima and the definition of the adjoint, and the last equality is just the definition of the norm [1] of the functional A*f. Now, since the range of Ej1 is contained in L¡ and the operator Е^/Це1!! has norm one we obtain the set containment {feX:f =(Efg)/||E,i, ||g||si}c{feX :feLi, M s 1} (18) 230 IAEA-SMR-11/ЗО and hence Eq. (16) becomes -i- + d j(A )S m axim um ||A *f || й m axim um { ||A>:'E^g|¡ / Це ^Ц} 1 llgll s i = [ || A *E f ||/ ||e +|| ] = [ IIA*( 1 -E f )II /¡Efll ] = [ II( 1 -E, )а ||/||е ^||] (19) = [|| A -E j a || / ||Ef|| ] Finally, upon combining Eqs (15) and (19), we have A-E í AH s||EiJ'||(di(A )+I|-) S 2{Pj (X)+ 1} (d,(A) + T|-) (20) 1 1 . 2 , 2Pi (X) + 2s. (A) + 2d. (A) S 4s. (A) + 2 - + —: i2 i2 ( •) Hence if A is s-compact the s¡ (A) converge to zero implying that the sequence of finite-rank operators E¡ A converges in the uniform operator topology to A, completing the proof of the theorem. The theorem has a number of immediate corollaries where under appropriate assumptions a class of operators may be shown to be s- compact and thus have the approximation property. In particular, since Sp£ S„we immediately obtain: Corollary: Every operator of class Sp, U p S a has the approximation property. Although the Dp classes are not, in general, s-compact it is possible to show that the operators of class Dj are s-compact for if dj(A) is an sequence then id¡(A) goes to zero and since p¡(Z) § i we have s¡ (A) = Pi(A)di(AMid.(A) (21) which converges to zero as i goes to infinity. Hence s¡(A) is a Q sequence if d¡(A) is an sequence and we have: Corollary: Every d-nuclear operator has the approximation property. Finally, if the p¡(X) sequence is bounded the every compact operator is s-compact since the d¡(A) go to zero for compact operators [4, 5] and we have: Corollary: Every Banach space X with a bounded p¡(X) sequence has the approximation property. Clearly, the corollary yields the classical result that every Hilbert space has the approximation property [2]. GOLDSTEIN and SAEKS 231 5. O P E R A T O R ID EA LS As with their Hilbert space antecedents [2], the trace class operators Sp and Dp form ideals in the algebra of bounded operators on a Banach space. For this purpose we equip Dp with the norm 1 /p 1 S p < oo (22) i = о and Sp with the norm Г .°° i/p Liip = Y,Si(A)P 1 S p S oo (23) 1 = 0 while Dœ and S are normed by IIAjj = maximum {d¡(A)} = d0(A) = ||A|| (24) °° о si -= «> and IA IL = m axim um { s ; (A)} (25) Os i respectively. We then have: Theorem: The trace classes Sp and Dp, I S p S » of operators on a Banach space X are normed by || ||p and || ||p, respectively, and are closed two-sided ideals in the algebra of all bounded linear operators on X. The theorem results from standard techniques, analogous to those used on Hilbert space [2, 3], and its proof will therefore not be given here. Similarly, techniques analogous to those used in Hilbert space yield characterizations of the duals of the Sp and Dp classes and most of the other properties which one associates with trace class operators [2,3]. As such, the Sp and Dj, classes are, indeed, natural generalizations to Banach space of the classical trace class operators. 6. CONCLUSIONS Our purpose in the preceding has been to indicate one method by which the structure theory for compact operators on a Hilbert space can be extended to Banach space. The key to the method is the use of the minimax principle, rather than eigenvalues, as the elementary concept. This yields a viable generalization, though one which is not spectral theoretic in nature, which naturally extends much of the Hilbert space theory to Banach space. In particular, it has been shown that: i) an s-compact operator. A, has the approximation property; ii) the p-trace class operators form closed, two-sided ideals in the algebra of bounded operators. 232 IAEA-SMR-11/3 О Unlike the above results, a satisfactory generalization to Banach space of the Schmidt representation has yet to be found though we conjecture that such a representation exists for s-compact operators in the form ( 26) i = 0 where the are, after appropriate normalization, the extremal vectors ф. which maximize ||Аф. - L.|| and 0. = Аф.. Similarly, there are a number of open problems associated with the approximation property. In particular, s-compactness is a necessary condition for an operator on a Banach space to have the approximation property and the condition that p¿(X) be bounded is necessary for X to have the approximation property. REFERENCES [1] RIESZ, F. , SZ.-NAGY, B. , Functional Analysis, New York, Ungar(1955). [2] GOHBERG, I.C ., KREIN, M .G ., Introduction to the Theory of Linear Nonself-adjoint Operators, Providence, Am. Math. Soc. (1969). [3] DUNFORD, N. , SCHWARTZ, J. , Linear Operators, 2, New York, Wiley (1963). [4] LORENTZ, G .G ., Approximation of Functions, New York, Holt, Rinehart and Winston (1966). [5] GOLDSTEIN, R. A ., SAEKS, R. , "Trace class, widths and the finite approximation property”, Bull. Am. Math. Soc. (to appear). [6] KAKUTANI, S. , "Some Characterizations of Euclidean Space”, Proc. Math. Soc. Japan 16 ( 1939) 93. [7] GOLDBERG, S., Unbounded Linear Operators, New York, McGraw-Hill (1966). [8] MURRAY, F.J., "On complementary manifolds and projections in spaces Lp and Í Trans. Am. Math. Soc. 40 (1937). [9] ENFLO, P., unpublished notes. [10] HOLMES, R. B., A Cours in Optimization and Best Approximation, New York, Springer-Verlag (1972). [11] LUENBERGER, D .G ., Optimization by Vector Space Methods, New York, Wiley (1969). IA EA-SMR-11/31 RIGIDITY AND ENERGY R .A . GOLDSTEIN, P.J. RYAN Department of Mathematics, University of Notre Dame, Notre Dame, Ind. , United States of America Abstract RIGIDITY AND ENERGY. Various concepts of rigidity are presented and illustrated. Among other items, infinitesimal rigidity is formulated globally and the differential equation for an infinitesimal isometric deformation is given. 1. RIGIDITY If you picture a Riemannian manifold, you are seeing it immersed in some Euclidean space. Two questions naturally arise. First, given a Riemannian manifold M, can it be isometrically immersed in some particular Euclidean space? This is a question of existence of a map with specific properties. In this paper, we are interested in the other question — uniqueness of this map when it exists. In other words, when you picture a certain Riemannian manifold M, is what you see the only possible picture, say, up to an isometry of the Euclidean space? Definition: A Riemannian manifold M which can be isometrically immersed in En is said to be rigid in En if for any two isometric immersions г: and r2, there is an isometry 233 234 GOLDSTEIN and RYAN 2. CONTINUOUS RIGIDITY In this section, we discuss a second type of rigidity more consistent with the dictionary meaning of the word, i.e. resistance to bending. We call this concept continuous rigidity. Definition: Let S = (M, r) be a submanifold of a Riemannian manifold M. Thus r: M -* M is an immersion. Let I = [-1, 1] . A map 7 : I X M - M is a deformation of S if 7 0 = r and 7t is an immersion for each t. Here 7t (x) = 7 (t, x). Each immersion 7 t induces a Riemannian metric gt on M and each closed curve on M has a length L(t) measured by the metric g. Definition: Let 7 be a deformation of S. Then 7 is an isometric deformation (ID) of S if gt = g0for each t. Definition: Let S = (M, r) be a submanifold of a Riemannian manifold M. Then S is^said to be continuously rigid if for every ID of S there is a curve 7 t (x) = Clearly, S is continuously rigid if M is rigid. On the other hand, Cohn-Vossen's example is continuously rigid but not rigid. The following conjecture is still open: Every closed surface in E 3 is continuously rigid. 3. INFINITESIMAL RIGIDITY We now come to the linearized version of rigidity (a notion used in elasticity theory) called infinitesimal rigidity. Roughly speaking, the infinitesimal theory is just the continuous theory with term s of second and higher order neglected. For example, if S is a surface in E 3 the analogue of an ID is an infinitesimal isometric deformation (IID) which is a deformation 7 with line elements satisfying (dYt )2 = (d T0)2 + O (t2) More specifically, in the notation of the last section, we have Definition: A deformation 7 is an HD if g'(0) = 0. Here, we are regarding gt as a curve in the finite-dimensional vector space of (0, 2) tensors at a point of M. Note that this also means that L'(0) = 0 for each closed curve on M. Definition: A submanifold S = (M, r) of E n is infinitesimallÿ rigid (IR) if for every IID 7 , there exists a curve 7t (x) = For any deformation y , we can write Tt = 7o+ t z + 0 ( t 2) (2) If Yt - cp(t) о r, then it is easy to verify that z = a r + b (3) for some constant skew-symmetric matrix a and constant vector b. One can also verify that a submanifold is IR if and only if the vector z determined by (2) is of the form (3). The best known result on infinitesimal rigidity is the following theorem due to Liebmann. Theorem: Every closed surface in E 3 with K ê О (К not identically zero on any open set) is infinitesimally rigid. In particular, if К > 0 the conclusion hold s. We see that the hypothesis К i 0 is not sufficient for infinitesimal rigidity of closed surfaces by a modification of the next example. Example. Let S = (M, r) be the open unit disk in E 2 with r the inclusion map into E 3. Let 7t (x,y) = (x, y, t(l-x 2-y2)) (te[ - 1 , 1 ] ) Then у is an IID but not an ID. Furthermore, the corresponding z cannot come from a Euclidean motion. Thus S is not infinitesimally rigid. 4. GLOBAL FORMULATION OF INFINITESIMAL RIGIDITY Let S = (M, r) be a submanifold of M. Let E be the restriction to M of the tangent bundle T(M). The set of sections of E is denoted by Г(Е) and such a section is called a vector field along r. E becomes a Riemannian vector bundle by restricting the metric of M to the fibres of E. If V is the connection on M, one defines a connection D on E by (Dx u) (p) = (Vxu) (r(p)) where Xe x(M), и£Г(Е), peM and X and u are vector fields extending r„X and u, respectively, on some neighbourhood of r(p). For иеГ(Е), the exterior derivative du of u is the E-valued 1-form defined by (du) (X) = Dxu A lso , if в is an E-valued 1-form, then E-valued 2-form dg is defined by d в (X, Y) = Dx (6 Y) - Dy (0X) - 0 [ X, Y] We note the relationship d (du) (X, Y) = R (X, Y) u 2 3 6 GOLDSTEIN and RYAN where R is the curvature tensor of M. Furthermore, if M is Euclidean, we have d2u = 0 for all uer(E). Definition: If 7 is a deformation of S, then the deformation vector field z is the section of E whose value at x is the initial tangent vector to the curve t - 7 t (x) Thus zx is the "initial velocity" of x under the deformation. T h eo rem : 7 is an IID if and only if for all X, Y in *(M), The E-valued 1-form p = dz is called the rotation form. Note that 7 is an IID if and only if < pX, Y > + < X, pY > =0 We restate the definition of infinitesimal rigidity since we are now allowing M to be non-Euclidean. Definition: S is IR if for each zGF(E) satisfying (4), there is a cúrve i/>(t)eI(M) such that the map (t,x) - is a deformation of S with deformation vector z. In such a case, we say that z is triv ia l. Remark: When M is Euclidean, z is trivial if and only if it is of the form (3). 5. EXAMPLES AND COUNTEREXAMPLES In this section, we state several theorems concerning rigidity of hyper surfaces of higher dimension. Proofs may be found in Ref. [ 3] . Theorem: The hypersphere Sn(R) of radius R in E n+1 is IR. Theorem: If R Q < R, Sn(RQ) in S n+1 (R) is IR. Theorem: Any hypersurface of E n + 1 some open set of which lies in a hyper plane admits a non-trivial IID. Theorem: The great sphere Sn(R) in Sn + 1 (R) is not IR. This last result contrasts with the theorem of O'Neill-Stiel [ 6 ] which says that Sn(R) in S n + 1 (R) is rigid in the sense of section 1. lAEA-SMR-11/31 237 6. THE DIFFERENTIAL EQUATIONS FOR AN IID Let S = (M, r) be a submanifold of M. Suppose M and M have dimensions n and n+p, respectively. An orthonormal set {et}"ÍÇ of n+p sections of E is called an adapted frame on S if the first n sections are tangent to M. Proposition. Let {e;}"^Ç be an adapted frame and let be the coframe on M dual to |ej}"=1. Then as a section of E, Remark: To explain notation, if ser(E ) and a is a 1-form on M, then as is the E-valued 1-form defined by (as) (X) = a(X)s (Xe*(M )) From now on, we take n = 2 and M = E3. The ordinary cross product in IR3 defines a map Г (Е ) X Г (Е ) - Г(Е ) which extends to all E-valued forms. Now suppose z is a deformation vector field satisfying (4). The rotation form p = dz gives rise to a section РеГ(Е) called the rotation field such that for X ex (M), p X = P X X Lemma: dP has no normal component. Corollary: There exist 1-forms T j and t 2 on M such that dP = г1 е1 + т2 е2 (5) and Tia2 - T2CTi = 0 (6 ) Here juxtaposition of forms stands for the ordinary wedge product. \ Corollary: There exist functions a, J3 and у on M such that T1 _ a & al -a K- .У l°-2j Let ш be the connection form and let 0 -ш 238 GOLDSTEIN and RYAN Let cjj and u 2 be forms (essentially, the second fundamental form) defined by the equation de = Qe - u e3 U)i w here e = ei and u = Le 2- -U2- Recalling (5), we see that d2P = 0 implies dr - ftr = 0 (7) and t * u = 0 ( 8 ) It is often possible to choose a frame so that the vectors are principal and Eqs (7) and (8) reduce to a simpler form. Such is the case in the following example: Surfaces of Revolution: Consider the surface of revolution of the graph of a function g parametrized by r and 0 a,s follows: x = r cose у = r sin 0 z = g(r) A global principal adapted frame is given by ex = (- sin 0, cos 0, 0) e 2 = £ (cos 0, s in 0, p) e 3 = j? ( p c o s 0, p s i n 0, - 1 ) where Г 1 = n/ 1 + p2 and p = g' (r) (jj = rd 0 dcj = drd 0 erg = i _1dr dcr2 = 0 и = -Hr _1 Oj Wl = рш 102 = -P 1^3^ k j = - p i r "1 k2 = - p 'i3 к = -E-f (1 + p2) rp' H ere, k 2 and k 2 are principal curvatures. We denote their ratio kx/k2by k. Whenever {ex, e2}is a principal frame, we have Uj = kjffj and u)2 = k2a 2 In view of (8), we have the identity kj/3 - k 2 Y = ° IAEA-SMR-11/31 239 Thus, provided that к is well-defined, we can write (7) in the form -T df + (dT - fiT) f = 0 where a f = and T = °1 C2 A - ' CT2 kcjj 1 0 Multiplying by and setting 0 -1 A = °1 CT2 -ka we have -A df + (dA + ÍJA) f = 0 (9) which we abbreviate L f = 0 Now Lf is a 2 X 2 matrix of 2-forms. The infinitesimal-rigidity problem now boils down to the following question: Does the differential operator L = -Ad + (dA + fiA) admit non-zero solutions for Lf = 0? 7. INFINITESIMAL RIGIDITY WITH BOUNDARY CONDITIONS О Non-compact surfaces in E are seldom IR as such. It is natural in a physical situation to consider holding certain parts of a surface fixed while bending the remainder. Thus, we shall allow our surfaces to be oriented two-dimensional manifolds with boundaries contained in larger surfaces. All deformations are assumed to be defined on the larger manifolds. In this situation, an IID determines and is determined by a field z satisfying (4). Definition: Let S = (M, r) be a surface in the sense of the above paragraph. If M0C M, then S is IR mod M0if dz|M 0 = 0 implies z is trivial. Remark: In our applications, M0 is a portion of the boundary of M. Definition: In term s of the operators introduced in section 6 , we define the deformation energy of M to be the matrix of 2-forms к - dA + fiA + A fi 240 GOLDSTEIN and RYAN In case к is not defined everywhere by this formula, we set к = 0 on the remainder. Then we have the identity 2 "e ll and the matrix of functions Ф 1 - 02- _Ф2- The expression <( Af, OgMis evaluated sim ilarly. Definition: к is said to be positive on M if k = -1 IAEA-SMR-11/31 241 (Note that H = kx+ k 2 = mean curvature = 0, or the catenoid is a "minimal s u r f a c e .") dA = I dr d0 ч/rZ- 1 0 fi = -cr-, r 2 - 1 Afi = fiA = 0 -1 2s/ r 2 - 1 a l CT2 2ч/ r 2 - 1 1 - The determinant of this matrix is r.-4 (r2 - 2)2 which shows that к is positive. If we consider portions of the surface bounded by two curves of the form r = const, then (j A) (90) - Because of orientation, A will be positive on one boundary curve and negative on the other. Thus, if one boundary curve is held fixed during an IID, the corresponding solution to Lf = 0 must be trivial. One can sometimes replace a given problem for which к is not positive by an equivalent problem for which к is positive. Let a, b and X be real functions on M such that a 2 + kb2 f 0. Let -kb V = a Then we can define a new operator L y x by L v x g = L(e Vg) (И) and arrive at a new energy term kv x satisfying an equation analogous to Eq.(10). Then if x g Г 2 V _1f positivity of kVi ^ together with a boundary condition will imply g = 0 and hence f = 0. The following is an example of this technique: 242 GOLDSTEIN and RYAN Example: the paraboloid. The curve y = x2 is revolved around the y axis and S is the resulting surface of revolution. Using the framework of section 6, we have '0 -CTl A = CT1 n = — a 2 -ktjj r .CTi 0 -f i -! Oj 02 = £2 A 1 0 dA = - r 0 8r 2 J1 I -1 0 К = — r 0 8 ^ 2 CT1 °2 -2 -2 k2 =' (1 + 4 r 2)1/2 (1 + 4Г2 )3 /2 к = 1 + 4Г2 Thus к is not definite. But now set x g = e 2 f For this example, we shall be using X = - (3/2) lnr, but work with a general X for a while in order to better illustrate the method. Note that we have taken V = I in (11). In this case we suppress mention of V in our notation. More specifically, we have A h L f = L (e2 g) = e 2 (Lg - i A dX g) W riting L x g = Lg - I A d X g and = к - Ad X we deduce with a computation that 2 If it is known that L^g = 0, kx is positive and g = 0 on ЭМ, then g must be identically zero. For the paraboloid L^g = e"x/2 Lf = 0 IAEA-SMR-11/31 243 A lso , 3Í -AdX = <*i °2 dr 2r -ff2 -kcr, 1 0 -AdX = M 2r 0 - (1 + 4 r2) °1 <^2 Thus 0 CTj 02 Kx = 2r 1 + 4 r2 which is positive. Thus we have the following theorem: Theorem: The portion M of the paraboloid y = x 2 + z2 lying between two planes y = constant is IR mod 3M. Remark: By further modification of L^, one can force the boundary integral in (12) to be positive on one of the boundary curves. Thus only one of the boundary conditions is necessary. The following theorems can be proved using techniques sim ilar to those for the catenoid and paraboloid: Theorem: Every minimal surface whose flat points (if any) are isolated is IR mod ЭМ. Theorem: Let g be a function whose graph y = g(x) has only finitely many points of inflection. Then the surface of revolution of g about the y axis is IR m od 3M. Theorem: Let S be a developable surface whose flat points form a set of measure zero. Then S is IR mod ^ where is any closed curve transversal to the generators of S. REFERENCES [1] EFIMOV, N .V ., Qualitative problems in the theory of deformation of surfaces, Differential geometry and calculus of variations, AMS translations, Ser. 1, £, 274. [2] FRIEDRICHS, K. O ., Symmetric positive linear differential equations. Comm. Pure Appl. Math. 11 (1958) 333. [3] GOLDSTEIN, R.A., RYAN, P.I. , Infinitesimal rigidity of submanifolds (to appear). [4] GOLDSTEIN, R. A ., RYAN, P. J. , Infinitesimal rigidity and the energy method (to appear). [5] KUIPER, N. , On C1 isometric embeddings, Proc. Math. Acad. Sci. Ser. A 58 (1955) 545, 683. [6] O'NEILL, B., STIEL, E. , Isometric immersions of constant curvature manifolds, Mich. Math. J. 10 (1963) 335. [7] STOKER, J.J. , Differential Geometry, Wiley-Interscience, New York (1969). IAEA-SME-U/32 PHASE TRANSITIONS IN D-DIMENSIONAL ISING LATTICES R .A . GOLDSTEIN Department of Mathematics, J.J. KOZAK Department of Chemistry, University of Notre Dame, Notre Dame, Ind. , United States of America Abstract PHASE TRANSITIONS IN D-DIMENSIONAL ISING LATTICES. Under the assumption of its existence, the partition function of a D-dimensional Ising lattice is evaluated in various regimes of 6 = 1/kT, and possible phase transitions are characterized. 1. INTRODUCTION The fundamental problem of mechanics is that of characterizing the time evolution (or flow) of a dynamical system given the initial conditions. Recall that a dynamical system is a collection (M,ju,cpt), where M is a smooth manifold, ц a measure on M defined by a continuous positive density, and the "flow" eft : M -» M (t e IR) is a one-parameter group of measure - preserving diffeomorphisms [1]. (Locally, cpt is defined by a first-order system: x =f(x). ) If the system contains relatively few particles, then a complete description in term s of the Hamiltonian equations of motion 9H q = Эр ЭН 9q where (q, p )eR 6N, N is the number of particles and H = H(q, p) is the Hamiltonian, is, in principle, possible, and the flow can be determined by integrating the differential equations given the initial co-ordinates and momenta of the individual particles. On the other hand, if the system contains very many particles (e.g. Avogadro's number), then the above approach is really out of the question, and, as realized by Maxwell, Boltzmann, Gibbs and Poincaré, it is far more sensible to introduce a statistical approach for the study of complex dynamical systems. Whereas the statistical approach would seem to be forced upon us by our ignorance of the initial co-ordinates and momenta of 1023 particles, and the impossi bility of computing simultaneously the dynamical history for each of these particles, it is well to keep in mind the "point" of doing such a calculation. We are reminded of Schrôdinger's often-quoted remark [2]: "It dawns upon 245 246 GOLDSTEIN AND KOZAK us that the individual case is entirely devoid of interest, whether detailed information about it is available or not, whether the mathematical problem can be coped with or not. We realize that even if it could be done, we should have to follow up thousands of individual cases, and could make no better use of them than compound them into one statistical enunciation. The working of the statistical mechanism itself is what we are really interested in". This shift from a deterministic to a probabilistic point of view in the study of complex dynamical systems carries with it the impli cation that a contracted description of a many-body system, one that involves fewer than the 6N co-ordinate and position variables, is not only necessary but desirable. That such a contracted description, one in which a new set of variables is identified, can be constructed is, perhaps, not too surprising. What is astonishing, however, is that the entire body of information for a sy stem of 1023 particles can be summarized in terms of three or four variables (depending on the constraints imposed and the composition of the system)'. This contraction is known as thermodynamics, and it may be taken as the principal aim of statistical mechanics to attempt to show how it is that the macroscopic behaviour of matter, as systemized in the laws of thermodynamics, follows from (or can be understood in term s of) the microscopic laws of atomic theory. In recent years, an increasing number of mathematicians and physical scientists have turned their attention to an examination of the mathematical structure and underlying assumptions of equilibrium and nonequilibrium statistical mechanics for the excellent reason that there still remain fundamental problems which are, as yet, unsolved, this despite the fact that certain of these problems have been studied actively for over a century. To place these problems in their proper context, and to state the mathe matical questions involved precisely, it would, of course, be necessary to undertake a complete exposition of the theory, a task which is obviously unrealistic given the space limitations of this contribution. Rather, what we shall try to do here, is to pose one problem of current interest in statistical mechanics in some detail, the problem of phase transitions in D-dimensional Ising lattices, and then to indicate how a synthesis of bifurcation theory and the theory of asymptotics can lead to a preliminary understanding of the occurrence of singularities in the thermodynamic functions at certain, well-defined points in the temperature spectrum. We mention in passing that a fundamental problem in statistical mechanics, a problem which we shall not even consider in this paper, is the mathematical proof of the existence of a certain function which plays an essential role in the theory. This function, called the partition function Q, is understood as (apart from a normalization constant) the sum over all energy states Ej of the system under study, Q exp [ -/3 Ej ] j where 3 is a parameter related to the temperature via Boltzmann's constant, к (in particular, 3 = 1/kT). It turns out that all macroscopic thermodynamic variables (e. g. pressure, free energy, entropy, etc. ) can be expressed in terms of this function, and when physicists or chemists "do" statistical mechanics, a primary objective is to evaluate Q, given the classical or IAEA-SMR-11/32 247 quantum-mechanical laws describing the molecular motion and interactions of the particles comprising the system being studied. It is interesting to note, then, that the fundamental assumption of Boltzmann and Gibbs on which the existence of the partition function rests, the so-called ergodic hypothesis, was proved only recently by Sinai [3] for a system of N 3 hard spheres (that is, particles interacting via infinitely repulsive forces for r < a, where a is the diameter of the particle, and experiencing no forces beyond a) moving in a volume V with periodic boundary conditions. A proof that systems whose molecules interact via repulsive and attractive forces (i. e. systems in the "real world") are also metrically transitive has not yet been achieved! With respect to the present discussion, and for the class of systems considered in this paper, we shall assume the existence of the partition function, with our more modest objective being the evaluation (in some sense) of the partition function in different regimes of /3, and the characterization of possible (second-order) phase transitions in D-dimensional Ising lattices. Consider a lattice with a spin at each vertex of the lattice, and assume that each spin a can take on two values, +1 (spinup) and -1 (spin down). Clearly, if there are N vertices, there are a total of 2N possible con figurations of spins on the lattice; let us denote a particular configuration of these N spins by {a} . In the special case with which we shall be con cerned in this paper, i. e. the Ising model, the energy for one particular configuration of spins is i.J the prime on the first sum indicating that the sum is over nearest neighbours only. Here, J is the "coupling constant" between neighbouring spins, and H is the external magnetic field. Thus, the first sum on the right-hand side accounts for spin-spin interactions and the second term represents the interaction between the individual spins and the external magnetic field. In one dimension, the lattice becomes a chain of N spins, and N -1 N while,in two dimensions E { a } = - J i= i j= i 1=1j= i 1=1 j=l for a lattice with m rows and n columns, etc. We remark that the sub sequent analysis is facilitated if one adopts periodic boundary conditions. Accordingly, in one dimension the ends of the open chain are linked to form a ring; in two dimensions, the lattice is mapped on a torus, and in higher dimensions, sim ilar mappings of the lattice are constructed. 248 GOLDSTEIN AND KOZAK In physical term s, we note that since the interaction energy between two neighbouring spins is -J if both spins are up (or both down), and +J if one spin is up and the other down, one obtains a lower energy for a parallel configuration if J > 0. Therefore, the minimum energy for the parallel configuration is just E 0 = -JNq/2 -N|H¡ (periodic lattices) where q in this expression is the number of nearest-neighbour sites of a given site. Whereas in the presence of a magnetic field H > 0 the minimum energy is achieved for all spins parallel and up (with just the opposite configuration for H <0), in the absence of a magnetic field (H =0) the minimum energy is achieved either when all spins are up or all spins are down. The situation J > 0 corresponds to ferromagnetism, while J < 0 corresponds to antiferromagnetism (here one obtains a lower energy for antiparallel configurations). Finally, we note the existence of a quite different physical interpretation of the Ising model, namely, the lattice gas; here one assigns o. = +1 if a molecule is at the ith lattice site, and CTj = -1 if there is no molecule at the ith lattice site. This lattice-gas interpretation is quite useful since it allows the consideration of gas-liquid equilibria, and the study of phenomena near the critical point. The statistical-mechanical problem can now be posed: Given the en ergy E{ ct} for a particular lattice, evaluate the partition function where the sum over {a} denotes a sum over all possible configurations cfj = ±1 of N spins on the lattice. Now, what has been proved for this problem? The one-dimensional problem has been solved both in the presence and absence of a magnetic field [4]. The two-dimensional problem has been solved only in the zero-field case; this, of course, is the famous contribution of Onsager [5]. Nothing whatever is known about the exact mathematical solution to the three-dimensional problem, either in the presence or absence of a magnetic field. Our specific concern in this paper is to indicate an approximate method for obtaining the behaviour of the D- dimensional Ising model in the zero-field case. 2. THE ZERO-FIELD ISING MODEL Consider a regular lattice with a spin at each of the N sites and assume that the interaction energy between neighbouring spins may be written as -J ct. cr. where J(J ^.0) is the interaction energy, and each spin can take on the discrete values ±1. Then, in the absence of a magnetic field, the partition function can be written down at once; it is exp - (1) kT { °i = ±1} IAEA-SMR-11/32 249 Here, the set (a} denotes a particular configuration of spins, and S ' denotes a sum over nearest neighbours (only); the factor 2"N is included so that the partition function is normalized to unity. Suppose now that in computing the sum over nearest neighbours, we count each interaction twice; then, using the notation 2 to indicate this double counting, the partition function a ssu m e s the fo rm :l,) 0 J Qn = 2~N У exp (2 ) {°i = ±1} The point of counting each interaction twice is that the matrix corresponding to the quadratic form is symmetric. Because of this symmetry, a i.j diagonalization of the quadratic form is always possible (via an orthogonal transformation), and if, in addition, we assume periodic boundary conditions then the matrix of the coefficients will be of the cyclic form. For definite ness, let 3 denote the column vector: Berlin and Kac [6] show explicitly that a matrix M can be constructed such that N ^ a¡ о- =5r>l-M '< /= ^ ai Mij ffj (3) i.j i. j = 1 where the cyclic matrix M has the real, orthogonal eigenvectors V (normalized to unity): 2ir (k-l)(s-l) 27r (k-l)(s-l) cos (4) Vk = v ks N N and eigenvalues mp D m p = 2 I ) cos 27гр„ p =(p 1,p 2,...,p d) (5) u=l 0 S p â L - 1 Thus, the Ising partition function can be written in the form: Q = 2"n У exp| ^ a ' • M • a j- (6) h = ±1} where H = j3J. 250 GOLDSTEIN AND KOZAK A second simplification in the structure of QN can be introduced if one uses the following, linear algebraic generalization of the Gaussian integral[7]: exp = (2тг)~2 (det J exp П< '2 ^ k J ki ■|^Ук ( •Т’ 1)м У ^ + i^k k,{ k.fi к (7) This identity is valid for any set of n (real or complex) variables Çk and for any real, symmetric positive-definite matrix J. Indeed, this identify could be used immediately were it not for the fact that the cyclic matrix M need not be positive definite (the eigenvalues of M are cosines) though the matrix is real and symmetric. However, this difficulty is handled easily be adding, and then subtracting, a quantity a ( a > 2D) to the diagonal elements of M. Specifically, we write M = M + al -al ( 8) where I is the identity matrix, and then define W = M + a I (9) The matrix W is now a real , symmetric, positive definite matrix with eigenvalues ( ) “ p = m p + a 10 With the above identifications, the Gaussian identity can be used directly (set k real), and one obtains NH H Ч,=2Г«е^“ I 6XP W • a {o¡ = ± 1} - \ > ) (2тг)‘2 (det HW)-i N 1 — 1 y 1 • (HW)- exp Q dNy exp 1 *y SC & (И) l> k=l Since the matrix W appears everywhere as W 1 it is convenient to identify a new matrix —» —» A = W" (12) with eigenvalues P = (Pp P2< • • • - Pd) <13) 2irpv 0 - Pr tf-1 L LÏ . V = 1 IAEA-SMR-11/32 251 Then, after some manipulations, one obtains .n 4 , Г +°° P Qn = exp (2тгН) 2 (d etA )2 J . . .J exp { I(y)} dNy (14) where I(y) = -^| 3. ASYMPTOTICS The difficulties associated with evaluating QN exactly lead one to consider the possibility of obtaining an asymptotic estimate for the partition function. Let us focus on the integral appearing in the expression for Q¡^ : + « S d Ny exp{I(y)} (15) and attempt to construct log Sw Lim —N N - S (16) More precisely, we estimate SN by the method of steepest descent, where, in this problem, H appearing in the integral Sn plays the role of the largeness parametr [8]. In our approach, we begin by considering the behaviour of the integral, Eq. (15), in the neighbourhood of each of its critical points. We expand the integrand in a Taylor series about each of its critical points, and then sum up the contribution from each of those points. About one such critical point, say yC]., we write Ky) =I(yCj) + | < S y Cj, V2I(yCj) 6yCj > + . . . (17) where 6yCj = У - yCj Note that since yCj is a critical point, VI(yc.) = 0. Therefore, the con tribution to the integral I(y) at the critical point yc. is estimated to be yCj+A J e x p |l(y c.) + |-< 6y c.,V 2 I(yc.) 6yc. >|dNy (18) yC j- ¿ 2 5 2 GOLDSTEIN AND KOZAK where Д is a sm all neighbourhood about yCj.. If one proceeds to sum up the contribution from each of the other j critical points (j = 1 , 2, . . . , s), the net result is + 00 b +00 J. . .J exp{I(y)} dNy -If.../ e x p |l(y c.) + | < 6yc. , V2I(yc. ) 6yCj > | dNy -00 j = 1 “00 (19) where, relative to each critical point, we have extended the range of integration to (-00, +co), since the overwhelming contribution to the integral will come from the narrow range (yc. - Д, yc. + Д), with a negligible con tribution from anything outside this range. Now, the method of steepest descent works for maximum points, but in our approach we focus on critical points initially. Given these critical points, we proceed by transforming the path of integration into one of steepest descent, such that the critical points become maximum points. More explicitly, we transform the integral, Eq. (19), to a complex integral in the complex N-plane, by using Cauchy's integral formula. Consider the following co-ordinate transformation: yj _> giCïj (yj - yi. ) We then choose real coefficients ttj (j = 1 ,2 ,... ,N) such that V l(y)<0, a condition which gives the maximum points. The consequence is that one obtains the same result with yc <-» y 1 =0 corresponding to a maximum point on the path of steepest descent. Once the above change of co-ordinates has been made, we change co ordinates once again so as to diagonalize V2I(yc), and after performing some calculations, we obtain the following estimate for the free energy: ф И ( У с ,) ] -F(H) = lim = lim -i- log j exp --N*HH (det A ) ^ N->«> iN N -*■“ iN {det[ -V I(yc )]}’ j=i + 0 ( e ' a /H ) (H -0 ) (20) 4. DETERMINATION OF THE CRITICAL POINTS Having obtained an estimate for the partition function, Eq. (20), we must now determine the set of critical points of the function I(y). The critical points of I(y) are the points for which Э1(У) = 0 (i =1,2, . . . ,N) (21 ) 3y‘ y‘=y¿ or, У Ai] yJ + tanh y 1 = 0 (22) j=i У1 = Ус IAEA-SMR-11/32 253 Taking into account all of the y1, we obtain the operator equation A yc = H 6 (yc) (23) w here, — — - - Ус1 tanh y¿ Ус = Ус 6(УС) = tanh у I Усы tanh y^ Since tanh(O) = 0, we see at once that a particular solution to Eq. (23) for a ll H is Уо w here y0 is an N-component vector with one zero corresponding to each y ‘ (i = 1,2, . . . ,N). In fact, one can prove the following theorem: T heorem : F o r H < Aj = l / ( a + 2D), the minimum eigenvalue of A, the only critical point of I(y) is y0 = 0. Having identified the single critical point relative to the range H< Xj we now seek to determine the set of critical points for H >Xj. This amounts to finding the other solutions of Eq. (23); that is, we determine whether new solutions might arise from the known solution y0 for particular values of the parameter H. Phrased differently, we study the possibility that new solutions can bifurcate from the known solution y0. We recall that if bifurcation takes place, it must happen at one of the eigenvalues of the operator A. Accordingly, we write the general critical point yCv in a power series in a parameter e about the known critical point y0, and, similarly, we expand the strength parameter H in a power series about the value H0 (the latter to be determined). The perturbation scheme utilized below goes back to Euler, Poincaré and Rellich [9] Ус„ = Уо+ e У1 + е 2У2 + е3Уз+ • • • (24) Н = Но + е Hi + е2 Н2 + е3 Н3 - (25) We proceed by substituting these expansions into the operator equation A yc„-H tanh yCy= F(e) 254 GOLDSTEIN AND KOZAK and then, differentiating F(e) successively with respect to e, we set the result at each order to zero. Starting off, we write 9F(e) = 0 (27) de e = 0 from which we determine the relation A y1 = [H sech2 yc • Э6 yc + H£ tanh yc]£ = 0 (28) This leads to the result А У1 = H0 ya (29) This equation, which is just the variational equation corresponding to Eq. (23), defines a linear eigenvalue problem. We identify h 0 = x „ (x1 a 2 s...aBs...aN) (30) У1 = А„Ф„ (31) where the cp„ are the eigenvectors corresponding to the eigenvalues X^ of the operator A and A v is some constant, to be determined later. To continue, we construct 9 ^ ( 6 = 0 (32) Эе2 e= 0 which leads to the result ( A - X J y2 = 2 H j A v Ф„ (33) This equation has a solution if and only if the right-hand side is orthogonal to the null space of (A -X„), i. e. to the eigenfunctions of A corresponding to X„. But, the eigenfunctions of the homogeneous equation are (to within a constant) just ф„, i. e. (34) Hence, the right-hand side of Eq. (33) can never be orthogonal to the фи unless Hj = 0. Accordingly, from the second variation, we have Hj = 0 (35) У1 = A V % У2 = c„ A„ % (36) IAEA-SMR-11/32 255 w here cv is some constant. Finally, we construct 93F(e) (37) Эе3 € = 0 which leads to the result ( A - X J y 3 = ЗНзА^Ф^-гХ^ А3 Ф (38) w here, (€ )3 0 0 Ф = 0 (ф' )3 ... 0 (39) 0 0 Ф 3 and where 2TTÍV 2t i v 1 % = CO S + s i n (40) N . N J is the i-th component of the у-th eigenvector of A. Given the structure of the corresponding homogeneous equation, Eq. (38) has a solution iff the right-hand side is orthogonal to фу. This leads to the result (41) w here, IN № I D„ = 2 1=1 (42) i = 1 Given the eigenfunctions, Eq. (40), the coefficient Dy can be evaluated, and one finds that D„ =1. Hence. Ho = X„ At (43) Substitution of the results obtained thus far into the perturbation expansions for y„ and H leads to Ус„ = Ф„(е+е2) + 0 (е3) (44) H = X1/(l + e2) + O ie3) (45) where we have used that yQ = 0 and HQ = and taken advantage of the liberty in choosing cv to set cv = A v; finally, for simplicity, we replaced Aye by e. 256 GOLDSTEIN AND KOZAK 5. THE FREE ENERGY IN THE RANGE H < Relative to the range H < Xj, we had proved that there exists only one critical point of I(y), namely, y0 = 0. One can show quite easily that y0 = 0 is also a maximum point; this follows from the fact that the matrix ^ 2*(Уо) can be shown to be negative definite for the range H < X1# One can proceed, then, with the evaluation of the partition function, Eq.(20), and determine an expression for the free energy of the system via the relation -F = lim *°g Q n (46 N-»oO The result for the free energy is: 2тг D F = - . . . / dDu log 1 -2H ^ co s u1 (H -0 ) (47) 2 ( 2 т г ) 1 \ i = l where in the continuum limit we have let o'-*-2D and replaced Xx by A(w) where X(w) =■ D 0§uS2i (48) 2'I ) c o s u ‘ i = l We see that the asymptotic solution relative to the range H < хг (H -> 0) is exactly the Gaussian model of Berlin and Kac [6]. Our approach yields that in the limit N-»°o, H->0, the Ising model should behave like the corresponding D-dimensional Gaussian model. 6. THE FREE ENERGY IN THE RANGE H > XN In the range H >XN, it turns out that V2I(y0)>0; in other words, y0 is a minimum point for I(y) in this range. Thus, the Gaussian model yields the correct limiting behaviour of the Ising model for H -V2I(yc„) = | - sech2 (yCu) (49) This is achieved by determining the eigenvalues Uj and eigenvectors r¡¡ of the m atrix H S , where H S (yCl/) = A -H sech2 (yc¡1) (50) IAEA-SMR-11/32 257 We write Uj (e) = (51) k = 0 £ _ (52) k k! к = 0 Substituting these expressions into the operator equation H lim H = X„ lim Yr =0 0 e-» 0 v ЭН „ Ятг.» l u n — = 0 lim e-> о de e-*0 9 e Я2 lim —b i = 2ф„ f í = 2Xy e^o Эе one finds, using the same procedure as the one introduced in section 4, that the eigenvalues of the determinant H [-V2I(yc. )] are given by >1j (e) = (Xj-X„) + X„ e2[L„j-1] (54) where <55) Here, the cp¡ are the eigenfunctions and the Xj are the eigenvalues of the operator A, i. e. A cpj = Xj ф. Furthermore, we find that, to order e3, N N ft т =е2[Ьу>„-1]Д Xj П (l-Y-)+0(e3) (56) j = l j = l \*v V i y With the result, Eq. (57), the partition function in the range H > Xj assumes the form : N D 2 *jy Q n - H4 exp|l(yCw) l o g 1 - Xy a + 2^T cos } (57> " i* V Li = l X exp ^ - — NaH 258 GOLDSTEIN AND KOZAK We note that this expression for can be written in the alternate form: ( a - 2 D )-1 Qn = O(N) J àX exp {N E (X)} (58) (a+2D )_1 with D E(x> = - | * l s + 1 ^ - | l 0s (2X - H) - ¿ Д > ~J. dDu log 1 - x (a + 2 ^ c o s u y i=1 (59) where we have replaced the sum L by an integral over u, and replaced j * V the sum over X^ by an integral over X. This representation is achieved by application of Cauchy's integral formula and the residue theorem. Finally, the contour of integration was deformed such that the range of X jis (X , oo). Upon examining the above expressions (Eqs (58, 59)), we note that the integral in the exponent has a branch point at X =Xj where Xj =1/2D, and it is at this point where (exactly as Berlin and Kac show for the spherical model) that the possible phase transition takes place. Note, however, that we must still determine the value of H at which the transition occurs. Accordingly, in the next section we take up the problem of characterizing the possible singularities in the free energy, where the free energy is determined from the relation, Eq. (46). We proceed by using, once again, the method of steepest descent (though now in its classical version). 7. THERMODYNAMIC BEHAVIOUR IN THE RANGE H È 1/2 D, AND PHASE TRANSITIONS To evaluate the integral, Eq. (58), by the method of steepest descent, we need the maximum points of E(X); we recall that our analysis based on bifurcation theory gave only critical points, but the co-ordinate trans formation suggested in section 3 allows the identification of the maximum points on the path of steepest descent. Let us denote by Xs = XS(H) the path on which we are assured that E'(X$) =0, or, equivalently, é ' ¿ T r - ¿ K<2X> <60> while at the same time E"(Xs)<0, or equivalently, E " (X‘ ) = ( 2X T^-è K(2X) + ¿K'(2X)<0 (61) where, in the above expressions, 2l\ 1 K(2X) = (2^ I dDy ------iSD------(62) 0 l-\(a+2 V) C O S Ü J i= 1 IAEA-SMR-11/32 259 We remark that the Xs = XS(H) which satisfy Eqs (60) and (61) are the maximum points to be used in the calculation of the free energy via Eq. (46). For the purposes of the present discussion, we examine the behaviour of the heat capacity C„ = к H2(^|) (63) \ эн2/ эх Чэн/ ax2 on the path X^ = XS(H) to determine whether Cv exhibits a singularity, and if so, we shall characterize the behaviour in the neighbourhood of the singularity, and shall locate the point at which the phase transition takes p lace. In one dimension, E(XS) has a "potential" singularity when H->-§ and X —> However, by considering the behaviour of E"(XS) in the neighbour hood of the saddle point Xs, we find that lim E" (X ) = +00 i That is, the point (i, \) does not lie on the saddle curve [H,XS(H)]. Hence, we conclude that the free energy and all derivative properties are well behaved throughout the range H > X^, this because the "potential" singularity at H = •§-, X = ^ corresponds to a minimum of the integrand, and, con sequently, contributes negligibly to the thermodynamic behaviour of the system. It is interesting to note that although we do have bifurcation in one dimension, we do not have a phase transition [ 10]. In two dimensions, examination of the structure of K(2X) shows that the integral may have a singularity when 2X = | and allu^O . Again, we examined the behaviour of E"(XS) in the neighbourhood of the possible singularity at H = I, X = 1 to determine whether the point ( i , 1 ) lies on the saddle curve [H,XS(H)]. We find that lim E "(X S) = -oo H ^ i X - i In other words, a singularity in the thermodynamic heat capacity is achieved in two dimensions, since E"(XS) <0 corresponds to a maximum of the integral. Moreover, the behaviour of Cv in the neighbourhood of the singularity is found to be logarithmic in agreement with the behaviour found by Onsager in his exact solution of the two-dimensional Ising model. Note that whereas the singularity in X occurs at the bifurcation point, X=^, the singularity in H is shifted beyond the bifurcation point. The heat capacity is well behaved until the strength parameter H achieves a value of 0.25) a value which agrees favourably with the Onsager result, Hc = 0.44. Considering next the case of three dimensions, we again examine the singularity structure of E'(X) =0 along the saddle curve, Xs =XS(H). One possible singularity is the intersection of Xs = XS(H) with the line X= 1/2 H; 260 GOLDSTEIN AND KOZAK this intersection is found to occur at X = H =0, and hence at infinite temperature - a point which is never reached. The second possible singular point is at X = — , a point where the integral appearing in Eq. (59) may become singular. Here we find that lim E "(X S) = -oo H-» 0.12 K 12 That is, a singularity in the heat capacity is achieved when Xs -» ^2 and H -» Hc =0. 12 since E"(Xs)<0. We note that the nature of the singularity in the neighbourhood of the point (HC,XC) = (0.12, ^ ) is algebraic, behaving like (6 X - I )'1/.2 What we have not shown is the precise behaviour of Cv with respect to (H - Hc) (which amounts to solving for \ = XS(H)), and there fore we have not determined the critical exponent a. However, by con sidering XHh one shows in exactly the same manner as Berlin and Kac, that XHH has a jump discontinuity, that is, there exists a discontinuity in the slope of Cv at Hc. Finally, it is worth mentioning that the numerically estimated value of Hc for the three-dimensional Ising model is 0.22, and our critical temperature Hc = 0.12 is in fair agreement with this estim ate [1 1 ]. We mention in closing that an analysis sim ilar in spirit to the one presented above implies that for D s 4, the integral K(2XS) behaves like (X -XC)(D 1^ 2 log(X -Xc) even dimensions (X -X CP 2^2 odd dimensions Inspection of these results reveals that for D è 4 there is no longer any singularity in the specific heat,but the discontinuity in the slope of the specific heat still persists. For details on the above calculations, the reader may consult Ref. [12]. ACKNOWLEDGEMENTS The general approach taken here was suggested to the authors by Professor Robert W. Zwanzig of the University of Maryland. The authors wish to acknowledge several very helpful discussions with Professor Zwanzig. REFERENCES [ 1] ARNOLD, V. I . , AVEZ, A ., Ergodic Problems of C lassical Mechanics, W .A. Benjamin, In c., New York (1968). [2] SCHRÔDINGER, E. , Nature 153 (1944) 704. [3] SINAI, Ja.G ., in Statistical Mechanics: Foundations and Applications, W.A. Benjamin, Inc., New York (1967). [4] See, e.g ., THOMPSON, C .J., Mathematical Statistical Mechanics, The Macmillan Company, New York (1972). [5] ONSAGER, L ., Phys. Rev. 65 (1944) 117. IAEA-SMR-11/32 261 [6] BERLIN, Т.Н ., KAC, М ., Phys. Rev. 86 (1952) 821. [7] The continuum representation of the D-dimensional Ising model has been used by many authors. See, e .g . LANGER, J .S ., Ann. Phys. 41 (1967) 108. [8] For a discussion of this type of integral, see: DE BRUIJN, N .G ., Asymptotic Methods in Analysis, North Holland, Amsterdam (1961). [9] For a recent monograph on applications of bifurcation theory, see: KELLER, J.B ., ANTMAN, S., Bifurcation Theory and Nonlinear Eigenvalue Problems, W. A. Benjamin, Inc., New York (1968). [10] The absence of a correlation between bifurcation points and phase transitions in one dimension was also found in a study of the Kirkwood-Salsberg hierarchy of integral equations for a system of hard rods. See; ING-YIHS. CHENG, KOZAK, J .J . , J. Math. Phys. 14 (1973) 632. [11] See, e .g . FISHER, М ., Rept. Progr. Phys. 30 (1967) 615. [12] GOLDSTEIN, R. , KOZAK, J. , Physica 71 (1974) 267. IAEA- SM-11/33 DIFFERENTIAL CALCULUS IN LOCALLY CONVEX SPACES R. A. GRAFF Department of Mathematics, University of California, Berkeley, Calif., United States of America Abstract DIFFERENTIAL CALCULUS IN LOCALLY CONVEX SPACES. A class of locally convex spaces which the author calls differential spaces (D-spaces) is defined. The class includes all normed spaces, the nuclear spaces which commonly appear in distribution theory, and also each conjugate Banach space with the bounded weak-star topology. It is shown that the theory of Banach- space differential calculus can be extended with only slight modification to produce a theory of differentiable maps between D-spaces in which most of the basic results of Banach-space differential calculus admit generalizations. The fundamental results of the differential calculus of maps between finite-dimensional vector spaces are: (1) the closure of Ckmaps under composition, keN (2) the inverse function theorem (3) the existence of Ck flows for Ck vector fields (4) the existence, for each open subset U of IRn, of a smooth function f: IRn -» IR such that f(x) f 0'#=^-x e U (which im p lies the existen ce of smooth partitions of unity). The extension of differential calculus to Banach spaces has been completely successful, in that (1), (2), (3) are still true in the more general setting; and even (4) holds, at least for separable Hilbert spaces and certain other nice Banach spaces (due to J. Wells, unpublished). A theory of differential calculus for maps between objects in a category of spaces properly larger than the category of Banach spaces must, at least, satisfy (1) and reduce to the usual theory for maps between Banach spaces. The usual elementary results such as Taylor's theorem should generalize. However, the fundamental purpose of such a theory should be to provide a foundation for generalizations of (2) and (3) which are of use in the study of differential equations. In this paper, we shall outline a general theory of differential calculus, and define a category of spaces and maps for which it is easy to prove a generalization of (1). The definitions have the advantage of being intuitive and easy to work with, and the category of spaces is large enough to include C" (D", IR) and the space of distributions on Dn. Generalizations of (2) and (3) to a class of non-metrizable locally convex spaces which is important in global analysis also exist, but these will not be discussed here (see R efs [ 1, 2 ]). 263 264 GRAFF Many of the ideas in this article had their origins in paper [ 5] by Kijowski and Szczyrba. Let V and Z be locally convex spaces, U an open subset of V with 0 e U, and h: U -► Z a m apping of sets: 1. Definition, h is tangent to 0 at 0 if: (a) h(0) = 0. (b) for each semi-norm X on Z, there exists a semi-norm uonV such that: for each e > 0, there exists a neighbourhood Ue of 0 with Ue c U, such that y e Ue=>X (h(y)) й e ■ v (y). 2. Remarks, (a) h tangent to 0 at 0=^h is continuous at 0. (b) If V and Z are normed spaces, the above definition is equivalent to the usual condition U h(y) Il S II у II • a( Il у I with a(r) -» 0 a s r -* 0 3. Definition. Let U be an open subset of V, f: U -» Z a mapping of sets, and xeU , We say that f is (Fréchet) differentiable at x if there exists SL £ L(V, Z) such that rx is tangent to 0 at 0, where rx is the map defined by rx(y) = f(x + y) - f(x) - i(y). Recall that L(V, Z) is a locally convex space, and is given the topology of uniform convergence on bounded subsets of V. 4. Definition, f : U -» Z is C 1 if f is differentiable at each point in U, and if Df: U - L(V, Z) is continuous, f is Ck, к > 1, if Df is Ck_1. f is C°° if f is Ck fo r a ll к e N. 5. Remark. Note that, if f e L(V, Z), then f is C". If Z is a locally convex space, and X a continuous semi-norm on Z, let Nx = {xe Z: X(x) = 0}, so that is a closed subspace of Z. Define Ъ\ to be the normed space where underlying linear space is Z/N\, with the norm which X induces on this linear space. Let px : Z -» Zx be the natural projection. 6. Lemma. Let V and Z be locally convex spaces, U an open subset of V, f: U -Z, andxeU. Then: (a) f is differentiable at x >p\ ° f : U -» Z x is differentiable at x for each sem i-n o rm X on Z. (b) f is C 1< > p \ ° f is C 1 fo r each sem i-n o rm X on Z. The above lemma, while trivial to verify, has important implications: for it implies that, to study differentiable maps between locally convex spaces, it suffices to study the special case of differentiable maps with normed spaces as the range spaces. This is precisely the direction in which we shall now proceed. Let V be a locally convex space, E a normed space, and v a semi-norm on V. The semi-norm v induces a subadditive function V: L(V , E) -* IR+ u{oo} which is defined as follows: for each £ eL(V , E), we let v'(Æ) = sup { ||ü(v) || : i/(v) è 1}, w here ||jé(v ) || is the norm of ü(v) in the normed space E. Since i^is subadditive and non-negative, V "almost" makes L(V, E) into a normed space, except that some elements of L(V, E) might have "norm" equal to infinity. So we simply throw these elements away, and make the following definition: IAEA-SM-11/33 265 7. Definition. L^fV, E) is the normed linear space consisting of those elem en ts í e L(V, E) for which Щ£) < °o, with ~v as the norm function. 8. Remarks, (a) The linear inclusion L„(V, E) -» L(V, E) is continuous. (b) For each he L(V, E), there exists a semi-norm ц = ц(h) such that h e i y v , E). Note that the second part of the above remark implies that L(V, E) = U L„(V, E), where A is the set of continuous semi-norms on V. v e л 9. Lemma. L„(V, E) is canonically isomorphic to L(V„, E). Proof. Each functional in Ij(4 v, E) obviously induces an element of L„(V, E), and the topology on both spaces is induced by a norm: the topology of uniform convergence on the unit у-ball of V. So to prove the lemma, it suffices to show that each element of L„(V, E) induces an element of L(Vi/, E). Let £ e L„(V, E) and suppose x is an element of V such that v(x) = 0. If we show that i(x) = 0, then it follows that SL corresponds to an element of L(Vi/, E). Now, since i e L„(V, E), there exists n e N such that j| jC (v ) || S n fo r a ll v e V with v(v) S 1. But y(rx) = 0 for all r 6 IR, which implies that r ||i(x) || È n fo r all r e IR, which implies that ||i(x)|| = 0. Thus i(x) = 0. It is now possible to give a tentative definition of a class of locally convex spaces in which the study of differential calculus is fairly simple. 10. Definition. A Dj-space is a locally convex space V with the following property: if U is a neighborhood of 0 in V, E a normed space, and f : U -» L(V, E) a continuous map, then there exists a neighbourhood W of 0 with ff £U, and a semi-norm v on V, such that f(W) çL„(V , E) and such that the map f : W -» L„(V, E) is continuous. The obvious examples of Dj-spaces are the normed spaces. There exist other, non-trivial examples of Di-spaces, but we will defer discussion of these until we have established a few elementary results about differentiable maps between Dj-spaces. 11. Theorem. Let V and Y be Di-spaces, E a normed space, U open in V, W open in Y, f : U -* W a C 1 map, and g:W-> E a C' map. Then g о f is С 1. Proof. Note that D(gof) exists for each xeU , and D(gof)(x) = Dg(f(x)) • Df(x). Thus we need merely show that D(gof) is continuous. To show this, it suffices to show that, for each x e U, there exists a neighbourhood A of x in U such that D(gof) is continuous on A. So let xeU , By the definition of a Dj-space, there exists a neighbourhood В of g(x) and a semi-norm ц on Y such that Dg(B) с L M ( Y, E) and such that Dg : В -* LM ( Y, E) is continuous. Similarly, there exists a neighbourhood A of x (we may assume Ac f_1 (B)), and a semi-norm v on V, such that Díp^o f)(A) с L U(V, Y and such that D(pjjof): A -» L¡,(V, Yjj) = L(V„,Y(j) is continuous. Then, on A, we have the following factorization of D(gof); A D(pM° f) -— g )° f - L (V y, Yjj) x L(Y„, E) - L(V„, E) = L„(V, E) - L(V, E). The bilinear map L(V„, YM ) x L(YM, E) -» L(Vv, E) is C“ since V„ , Уц and E are all normed spaces, and the inclusion L v(V, E) -» L(V, E) is smooth, since 2 6 6 GRAFF it is linear. Since D(pMo f) : A -> L(V„, Y¡¡) and (Dg)of : A -» Ь(УД, E) are both continuous, we conclude that D(gof) is continuous on A. 12. Theorem. Let V and Y be D^-spaces, Z a locally convex space, U open in V, W open in Y, f : U -> W a C 1 map, and g: W - Z a C 1 map. Then gof is C 1. P ro o f. By p art (b) of L em m a 6, it suffices to show that p\o(gof) is C 1 for each semi-norm X on Z. But px°(gof) = (pxog)of. Now, since g is C1, L em m a 6 implies that p\og is C1. Thus (p\og)of is C 1 by the previous theorem . One other result which we shall state is a converse to the Fundamental Theorem of Calculus, which is essential to the proof of Theorem 19. 13. Lemma. Let V be a Di-space, Z a complete locally convex space, U a convex open subset of V, and f : U -» Z. Assume there exists a continuous map g : U -» L(V, Z) such that f(y) - f(x) = J g(ty + (1 - t)x) (y - x) dt for each x, y e U о Then f is C1, and Df = g. The essential ideas of this theory of differential calculus are contained in the C 1 theory discussed above. With this theory as motivation, we now give the definitions which we need to develop the Ck theory. Let V be a locally convex space, E a normed space, r e N, and v a se m i norm on V. Then v induces a subadditive map 7 : L C(V, E) -» IR+ c{oo} by 'v(i) = sup SL (v j, . . ., vr ) (I : v (v¡ ) S 1, 1 s i S r 14. Definition. LÍ,(V, E) is the normed linear space consisting of those elements i e L r(V, E) for which ’v(l) < oo, with V as the norm function. 15. Definition. A D-space V is a locally convex space with the following property: if U is a neighbourhood of 0 in V, E a normed space, and f : U -* L r(V, E) a continuous map, then there exists a neighbourhood W of 0 with W cu, and a semi-norm v on V, such that f(W) ç L jl V , E) and such that the map f : W -* Lr¡,(V, E) is continuous. One immediate consequence of the above definition is the following: 16. Lemma. If V is a D-space, Z a locally convex space, and r e N, then L(V , L r (V, Z)) = L r+1 (V, Z). 17. Remark. The above lemma implies that L (V, L(V, . . ., L(V, Z)). . . . ) = Lr(V, Z). Thus, if f : V -* Z is a Ck map, then D¿f is a map from V to L ‘(V, Z) for each 1 â i s k. 18. Theorem. Let V be a D-space, Z a locally convex space, U an open subset of V, f : U -* Z, and к e N. Then f is Ck"£=^p\of is Ck for each sem i norm X on Z. Proof. Using Remark 17, the proof is analogous to the proof of Lemma 6. IAEA- SM-11/33 267 The essential difference between the theory of differential calculus in the category of D-spaces and the theory in the category of D]-spaces is the following theorem, which follows immediately from Lemma 13, Remark 17, and the definition of a D-space: 19. Theorem. Let V be a D-space, E a normed space, U a neighbourhood of 0 in V, r and к non-negative integers, and f : U -» L r(V, E) a Ck map. Then there exists a neighbourhood W of 0 with W £U, and a semi-norm v on V, such that f(W) Ç L r„(V, E) and such that the map f : W -» L^(V, E) is Ck. 20. Theorem. Let V and Y be D-spaces, E a normed space, U open in V, W open in Y, and к e N. Assume f : U -» W and g : W -* E are Ck maps. Then g o f is C k. Proof. By induction: we know the result for к = 1. So let к > 1, and assume the result proved for the case (к - 1). Using Theorem 19, and the factoriza tion of D(gof) used in the proof of Theorem 11, it follows that D(gof) is Ck_1, and hence that gof is Ck. 21. Theorem. Let V and Y be D-spaces, Z a locally convex space, U open in V, W open in Y, and к eN. Assume f : U -> W and g: W -* Z are Ck maps. Then gof is Ck. Proof. Immediate from Theorems 18 and 20. It is now possible to prove other basic results about differentiable maps. For instance, if V and Z are locally convex spaces, U an open subset of V, f : U - L r(V, V), and g: U - L(V, Z), define the map g - f : U - Lr(V, Z) by (g- f)(x)(w) = g(x)(f(x)(w)) for all w e Vr, x e U. 22. T h eo rem . L e t V be a D -sp a ce. A ssu m e that f : U -» L r(V, V) and g : U -» L (V, Z) are both Ck maps. Then g • f is С k. The above theorem is useful in the study of inverse function theorems in the category of D-spaces. However, we shall devote the remainder of this article to showing that many of the non-normable function spaces commonly encountered in analysis are D-spaces. 23. Definition. An exponential space V is a locally convex space such that Vr is compactly generated for all r e N. 24. Remark. Note that a metrizable locally convex space is an exponential sp ace. Let Zi and Z 2 be locally convex spaces. For each r e N we define ev: Lr(Zi, Z 2) x Zi -> Z 2 to be the evaluation map, i. e. ev(f, w) = f(w) for all f e L '( Z 1 ,Z 2), w eZ'j, The following lemma is the fundamental result about exponential spaces: 25. Lemma. Let V be an exponential space, Z a locally convex space, U an open subset of V, reN , and f : U - Lr(V, Z) a continuous map. Then evo(f x I): U x Vr -* Z is continuous. 26. Definition. A Schwartz space V is a locally convex space such that, for each semi-norm X on V, there exists a semi-norm у on V such that v ê X and such that the induced map is precompact. 2 6 8 GRAFF The definition of a Schwartz space is due to Grothendieck, and his original article [3] remains one of the best references on the subject (another very readable reference is Ref. [4]). Our last result will be that an exponential Schwartz space is a D-space. Before giving the proof, however, we shall list some examples of exponential Schwartz spaces, and give some definitions. 2 7. E x am p les, (a) C “ (M, IR), the sp ace of sm ooth functions on any m anifold (compact or non-compact, with or without boundary). (b) the space of testing functions on any open subset of IRn. (c) H'~(M, IR), the space of distributions on any compact manifold M. (d) 1R“. (e) Let В be a Banach space. Then B' with the bounded weak-star topology is an exponential Schwartz space (differential calculus in this cate gory of spaces is studied extensively in Refs [1, 2]). 2 8. Definition. Let Zi and Z 2 be locally convex spaces, and let r e N. (a) Ls(Zj, Z 2) is the linear space of r-linear maps from Zi to Z 2 with the topology of pointwise convergence. (b) L^Z-l, Z2) is the linear space of r-linear maps from Zj to Z 2 with the topology of uniform convergence on precompact subsets of Zi. The above function spaces will be used in the proof that an exponential Schwartz space is a D-space. We shall only be interested in the case when Z x and Z 2 are normed spaces. In this case we have the following lemma, which is an immediate consequence of the definition of a precompact set: 29. Lemma. Let Ei and E 2 be normed spaces, let r e N, and assume that A is a bounded subset of Lr(Ei, E2). Then the subspace topology induced on A by LjiEx, E2) coincides with the subspace topology induced on A by Lc(E1; E2). 30. Theorem. Let V be an exponential Schwartz space. Then V is a D-space. Proof. Let E be a normed space, U a neighbourhood of 0 in V, reN , and f : U -* Lr (V, E) a continuous map. We must find a neighbourhood W of 0 and a semi-norm v on V such that f(W) с L£(V, E) and such that the map f : W -> L^(V, E) is continuous. By Lemma 25, the induced map g = evo(f x I) : U x V 1 -» E is continuous. Since g is continuous, and g(0, 0) = 0, there exists a neighbourhood W of 0 and a semi-norm X on V such that Il g(x, w) II S i for all x e W, and for all w = (vj, ,,,, vr)eV r such that X(v¡) S 1 fo r all 1 S i S r. Thus, f(W )cL\(V,E) = Lr (Vx, E). Unfortunately, f might not map W continuously into L r(Vx, E). However, since g is continuous, the map gw: W -* E which is defined by W-^— L r(Vx , E ) - ^ E x ---- - f(x )----- f(x)(w) is continuous for each w e Vx, which implies that the map f : W -» LjCV^, E) is continuous. Note that'Tfffy)) S 1 for all x eW, so that the topology on f(W) a s a su b sp ace of LljCVx, E) is the sam e a s the topology which f(W) inherits as a subspace of Lç(V\, E). Thus f : W -> Lç(V\, E) is continuous. IAEA-SMR-11/33 2 6 9 Finally, choose a semi-norm von V, i/U , such that the induced map V v -> V\ is precompact, which implies that the induced map Lc(V\, E) -* ЬГ(У„, E) = L r„(V, E) is continuous. Then f(W) с L r„(V, E), and f : W -> L lv(4, E) is continuous. This concludes our discussion of elementary differential calculus. For a more complete treatment, and applications to global analysis, see R efs [ 1, 2 ]. REFERENCES [1] GRAFF, R., Elements of local non-linear functional analysis, Ph. D. Thesis, Princeton University, Princeton, N.J- (1972). [2] GRAFF, R., Elements of non-linear functional analysis, in preparation. [3] GROTHENDIECK, A ., "Sur les espaces (F) et (DF)", Summa Brasiliensis 3 (1954) 57. [4] HORVATH, J., Topological Vector Spaces and Distributions, Addison-Wesley, Reading, Mass. (1966). [5] KIJOWSKI, J ., SZCZYRBA, W ., On differentiability in an im portant class of locally convex spaces, Studia Math. 30 (1968) 247. IAEA-SMR-11/34 SINGULARITIES IN "SOAP BUBBLES" AND "SOAP FILMS" Jean E. TAYLOR Department of Mathematics, Rutgers University, New Brunswick, N. J ., United States of America Abstract SINGULARITIES IN "SOAP BUBBLES" AND "SOAP FILMS''. The geometry of some surfaces arising in the calculus of variations and physically represented by soap bubbles and soap films is discussed. In this paper, we discuss the geometry of some surfaces which arise in the calculus of variations, surfaces such as are represented physically by soap bubbles and soap films. There has been an extensive literature on the structure of soap bubbles and films, notably including Plateau's two volumes on the subject [ 1 ]; however, it is only recently that the surfaces which must be considered in the appropriate minimizing problem have been given a thoroughly rigorous mathematical formulation. Before considering the mathematical model, let us look at the experi mental observations of actual soap bubbles. In Fig. 1 there is a sketch of three soap bubbles sticking together. If one looks at such a compound bubble, one notes that it consists of several (six here) "smooth surfaces" each having constant mean curvature, together with several (four here) "smooth arcs" along which three of these surfaces meet at 120° angles — that is, along these arcs the surface looks tangentially like Fig. 2 — and several (two here) isolated "points" at which four such singular arcs meet, bringing together six smooth surfaces, as in Fig. 3. That these are the only phenomena which occur stably (that is, which persist under small perturbations such as shaking and blowing) in any soap film or bubble is precisely Plateau's observation. A reasonable mathematical formulation of the problem which the physics of soap bubbles seem s to impose is that of enclosing constant volumes with the least possible total surface area, as has been recognized for many years. Precisely, let aj, ..., aN be numbers greater than zero, and let jd be the collection of all sets of N open sets (Aj, . . ., A N) with volume1 (A¡) = a ¡ for each i = 1, .. ., N. The problem is then to find (A1, . . ., A') e js' such that N Area( и boundary A!) i = i 1 1 Throughout this paper "volume", "area", and "length" mean precisely Hausdorff 3-, 2-, and 1-dimensional measure, respectively. Hausdorff measure agrees with any other reasonable definition of area (respectively, volume or length) on smooth submanifolds, but additionally gives a precise meaning to area when singularities may be present. 271 272 TAYLOR is a minimum over all jtf. If (A^, . . . , A^) gives such a minimum, then N (boundary A.1) will be called a "soap bubble". Similarly, given a compact set В of finite length, we define the closed set S to be a "soap film" with boundary contained in В if the area of S is less than or equal to the area of any surface S' obtained from S by a C" deformation of R3 which does not move B. Each of these formulations implies a variational formula. For a "soap bubble" S, it is that Area (S П B(p, r)) é (1 + Kr) Area (f(S n B(p, r))) for all p in S, all r less than some constant 6 larger than 0, and all deformations f such that {x: f(x) j- x} U {y: y = f(x) f x} is contained in B(p, r), where B(p, r) is the closed (three-dimensional) ball centred at p of radius r and К is a constant (which turns out to be dependent on the maximum mean curvature of the analytic part of S). For a "soap film", the corresponding inequality is the same except that we require К = 0 and r < dist(p, B). It has recently been shown that solutions to these problems do always exist and that the resulting "soap films" and "soap bubbles" are, in fact, except for a compact singular set of zero two-dimensional area, the union of two-dimensional analytic submanifolds of R3 [2]. We consider here the remaining set, the singular set. Specifically, we shall classify all the possible singularities, give the local combinatorial and differential structure of the surfaces near the singularities, and show that this classification, FIG. 1. A compound bubble. IAEA-SMR-11/34 273 FIG. 3. The cone T, restricted to the convex hull of its vertices. in fact, coincides with Plateau's axioms. (The work outlined in this paper is substantially contained in the author's doctoral dissertation [3] and will appear in detail later [ 4]. ) A nice property of area problems in the calculus of variations, as treated in the context of geometric measure theory, is that tangent cones exist at all points in solutions to the problem. A surface С is defined to be a tangent cone to a surface S at a point p if there exists a sequence of rad ii r ; -*• 0 such that С = lim £(l/r¡) т(р) (S П B(p, r¡)) i-> 00 where т(р) : R3 - R3, т(р)(х) = x - p £(l/r¡) : R3 - R3 £(l/r. )(x) = x/rt B(p, r) = R3 n íx: I x - p I s r} as before. 274 TAYLOR FIG.4. A spiral having every ray as a tangent cone. There may easily be more than one tangent cone at a point; for instance, consider the 1-dimensional spiral of Fig. 4. Conceivably, there might also not be any tangent cones at some point. However, if the surface satis fies the variational inequality above, then one can show that for sm all r r 2 e1^ . Area (T n B(p, r)) decreases monotonically as r decreases to zero. One therefore has an upper bound to the areas of the surfaces g[l/r) r[p) (T П B(p, r)) for small r; since the classes of surfaces one usually considers in geometric measure theory have strong compactness properties in the weak topology, for any sequence of radii decreasing to zero some subsequence of the corre sponding sequence of surfaces must converge. These tangent cones can be shown to have several properties: (1) They are cones (i.e. they consist of rays from their boundaries to the origin) (2) They are area minimizing (that is, Area(C) § Area (f(C)) for any deformation f: R3 -» R? which does not move ЭС). Therefore, to ask what tangent cones are possible is to ask "what are all the area-minimizing two-dimensional cones in R3? ". Since we are considering cones, it is sufficient just to characterize their boundaries. Two conditions on these arise almost immediately: (l)'that the boundaries consist of segments of great circles, and (2)' that these segments intersect IAEA-SMR-11/34 275 only 3 at a time at 120° angles. We note that, by the Gaass-Bon.net theorem, which gives the area of a face bounded by geodesics as 2 ir- L (exterior angles at vertices), the maximum number of vertices for any face outlined by geodesics as in (2)' above is five. The problem of finding all configurations satisfying (1 )' and (2)' above should therefore be soluble; and in fact the only possibilities are as follows: (1) A sin gle g re a t c irc le ; (2) 3 half circles at 120° angles (the cone over this boundary is precisely Fig. 2), (3) 6 segments of great circles forming the one-skeleton of a regular spherical tetrahedron (the cone over this boundary is as shown in Fig.3), (4) 12 segments forming the one skeleton of a spherical cube, (5) 9 segments forming the one-skeleton of a spherical prism over a regular triangle (only one such figure exists satisfying (1 )' and (2)' above), (6) 15 segments forming the one-skeleton of a spherical prism over a regular pentagon (again, there is only one of these), (7) 3 0 segments forming the one-skeleton of a regular spherical dodecahedron, (8) 24 segments forming 2 quadrilaterals and 8 pentagons, (9) 21 segments forming 3 quadrilaterals and 6 pentagons, (10) 18 segments forming 4 quadrilaterals and 4 pentagons. We now ask which of the above configurations actually satisfy the property that their cones are area minimizing; we see that (1), (2), and (3) do, whereas we can construct explicit deformations which save area com pared to the cones over (4), (5), (6), (7), and (10). To illustrate how the above is proved, we show here that the only configuration where all faces have four vertices is the regular-one (that is, where all edges have equal length), and that the cone over this configuration is not area-minimizing. FIG. 5. A spherical quadrilateral, extended. 276 TAYLOR Assume Q is a spherical quadrilateral and one side has length a . Take the two adjacent sides to a, and extend each of them (as geodesics) in both directions until they intersect (see Fig. 5). Now either two angles and the included side or 3 angles completely determine a spherical triangle; there fore given the one side length a, the angle of intersection of the extended geodesics is determined, the other angle of intersection must be the same, and therefore the side of the quadrilateral opposite to the side of length a must be of length a also. We further see that the two other sides must be of equal length, and that that length is determined by a . Therefore there is only a one dimensional family of quadrilaterals on the sphere (with 120° angles), and they are all rectangles. Now consider three adjacent such quadrilaterals, as in Fig. 6 . If one of them has side lengths a and ¡3, then the third edge 7 intersecting these two must have length a from consideration of the second quadrilateral and /3 from consideration of the third. Therefore, a = ¡3 = 7 , and all sides are equal (i.e. the configuration is that of the one-skeleton of a spherical cube). The cone over the one skeleton of a cube is not area-minimizing, as can be seen by dipping a wire frame formed into such a boundary in a soap solution. The result will be as diagrammed in Fig. 7; such a surface can in fact be obtained from a deformation of R3 applied to the cone. However, the simpler deformation which just makes the central surface a small square of side d can be shown itself to have area at least 0. 06d2 less than the area of the cone; therefore the cone is not area-minimizing. The results stated above can be combined into the following theorem. Theorem 1 [4]. The only tangent cones to "soap bubbles" and "soap films" are the disk, the cone Y of Fig. 2, and the cone T of Fig.3. Having this theorem, we have a grasp on the local structure of soap bubbles and films; however, as yet, we have given no indication whether behaviour of the type of Fig. 4 can occur. However, we can prove much more. IAEA-SMR-11/34 277 Theorem 2 [4]. Let S be a "soap film" or "soap bubble". Then (1) There are unique tangent cones at every point, (2) The points with Y as a tangent cone form a 1-dimensional Hôlder- continuously differentiable submanifold of R3 . (3) In a neighbourhood of a point with Y as a tangent cone, S consists of three two-dimensional Holder-continuously differentiable manifolds- with-boundary meeting at 120° angles along the submanifold of ( 2); in a neighbourhood of a point with tangent cone T, S consists of 6 H older- continuously differentiable manifolds with corners meeting tangentially as in Fig. 3. The main analytic estimate of the proof is an inequality relating the area of a surface within a ball to the length of its boundary on the boundary of the ball. The particular inequality proven is analogous to one studied by the late E.R . Reifenberg and termed by him an epiperimetric inequality. Reifenberg proved such an inequality when the tangent cones were assumed to be disks and the surface assumed to be area-minimizing (in general dimensions and codimensions, however) [5]; we prove here such an inequality for singular tangent cones (of dimension 2 in R3) to more general surfaces by a method totally different from Reifenberg's. Let us first define a parameter which monitors the behaviour of the surface at small radii. We already have that r "2 eKr Area (S П B(p, r)) decreases monotonically to a limit as r approaches zero; this limit is easily seen to be the area of a tangent cone at p. For simplicity, let us here only consider points with Y as a tangent cone; this number is then 3v/2. We therefore define, at such points p, ExcB (S, p, r) = r "2 eKr • Area (S П B(p, r)) - 3ir/2 Note that this quantity is always non-negative, by the monotonicity property stated above. 278 TAYLOR Using the notation p Ж W to denote the cone from a point p to a boundary W, we may now express the epiperimetric inequality we wish to prove as follow s: There exist к > 0, 6 > 0, e > 0, and J? < oo such that if 0 < E x c B (p X 8 (S П B(p, r)), p, r) < e and 0 < distance (g(l/r)Tp(pX3(SnB(p,r))), 0Y) < 6 for some SO(3), the group of orthogonal rotations of R3, then there exists a surface' Z obtained from S n B(p, r) by a deformation of R 3 and with 9Z = Э (S П B(p, r)), satisfying ExcB (Z, p, r) S (1 - k) ExcB (p Ж 9 (S П B(p, r))) + Í K r In other words, except for the term iK r, one can save area proportional to ExcB in the cone p X 9 (S П В (p, r)). If such an inequality can be proved, then by the existence of tangent cones the conditions are met for small enough radii (and, in fact, they are met uniformly for all p' with tangent cone Y in a neighbourhood of p). Since S satisfies the variational formula given before, we have for small rad ii E x c B (S, p, r) = r ' 2 e 1^1 Area(S n B(p, r)) - 37r/2 ê e2Kr Area(Z n B(p, r))r ’2 - 3тг/2 á (1 - к) [ Area(p X 9(SnB(p, r)))eKr r "2 - 3 7r / 2 ] +Î. Kr + 2Kr • Area(Z П B(p, r))r 2 S (1 - k) [ (r/2) [(d/dr) Area(S ПB(p, г))] еКг r ’ 2 - Зтг/2] + £'Kr [7, 4.2. 1] â (1 - к) [ (r/2) (d/dr)ExcB(S, p, r) + ExcB(S, p, r)] + Í 'K r This inequality can now be integrated in the appropriate regions to give (Зтг/2)г2 s eKr • Area(S П B(p, r)) s (Зтг/2)г2 (1 + A 1 r 2k/(1 ' k> ) where Ax is a constant which can easily be calculated. This upper bound to the growth in area of S near p limits the amount of twisting that can take place; one can in fact show that such a condition on the growth in area of a surface implies unique tangent cones and all the other conclusions of the theorem. (Reifenberg obtained conclusions analogous to (1) and (2) from his epiperimetric inequality but through a different proof [ 6 ] . ) The proof of such an epiperimetric inequality is thus the heart of the difficulty. It is done by contradiction; therefore there are no a priori estimates on k, the Holder exponent. The contradiction is as follows: F o r every v - 1, 2, . . . there exist numbers k„ > 0, >0, > 0, lv < oo, and r u > 0, and surfaces S„ with (1 ) lim„k„ = 1нп„е„ = lim^ó,, = 0, 11т„4„ = oo (2) ExcB(Sy,p ,rJ = ev IAEA-SMR-11/34 279 (3) d ist ML/r)Tjp)(p X 3(S„ ПВ(р , г Д Y) = 6V , and (4) ExcB (Z, p, r^) > (1 - k) e„ + for all Z with 3Z = 3(S„ n B(p, r„)). We m ay a lso assu m e (5) If lim ^e ;1 = 0, then lim^^r^eÿ 1 = 0 We may, in fact, assume (6) l i m ^ e " 1 = 0 since otherwise we immediately contradict the contradiction. The proof that the contradiction cannot be true is rather technical; one first succeeds in reducing it to the case where 3(S,riB(p, r v) c o n sists of a finite number of C1 arcs for all large v, and then to the case where it, in fact, consists of only three arcs. Both of these arguments involve showing one can push (by a deformation of R3) the given cone onto the cone over such a boundary, provided one adds a "patch" out at the boundary; in order for there not to be an immediate contradiction, the area of the "patch" must itself be insignificant in the limit and can be ignored. Once one is dealing just with sequences of cones over 3 arcs, one defines a new excess ExcyfS^) as follows. The cone over the three arcs, when translated so that its centre is at the origin and expanded to unit radius, must lie close to some orthogonal rotation of Y, by the distance hypothesis. One chooses the "best possible" such rotation and extends radially the cone over each single arc in turn until the boundary of the extension projects orthogonally, by the projection defined by the appropriate half plane of Y, into the unit circle in that plane centered at 0. One then defines 3 ExcY(Sy) = ^ [(Areaof extension of cone over ith arc) — i = 1 (Area of projection of this extension)] This new excess is proven to be related to the old in the following two ways: lim„ ExcY(S„) = 0 (easy) E x c B(Si;, P. r v) á ExCyiSJ (hard) Noting that the extended surface can be regarded as lying within the original cone by simply shrinking it by a factor of 1 / 2, one concludes that if the contradiction of the epiperimetric inequality holds for ExcB, it also holds (with different sequences k* , ó* , e* ) for Excy. But now we have surfaces lying over their projections, when we consider each arc independently, and we can apply the powerful methods of Almgren [ 2 ] to conclude that the surfaces must, in fact, be very close (i.e. within a constant tim es ExCytS^)1^2) to being the graphs of harmonic functions. The two conditions of being a cone and being very close to harmonic imply that the surface is very close to a plane; the three planes corresponding 280 TAYLOR to the three arcs cannot all be at 120° to each other because the ExcY has to come from somewhere. One shows that, at least, one of the angles between the planes must differ from 120° by at least a constant times Excy^,)1/2. It is then purely a geometric argument to see that if the angles differ from 120° by с • (Excy(Sb ))х^2 , then one can deform the planes near the centre line until they are at 120° angles and in so doing save area proportional to Excy(S^). But this gives us a final contradiction of the contradiction, and the epiperimetric inequality must therefore hold for some k> 0, 6 > 0, e > 0 and i< oo. / The proof of the corresponding inequality for points with T as a tangent cone is entirely analogous except that one reduces the problem to the case of six arcs rather than three. As we have seen already, such an inequality is sufficient to prove Theorem 2. In fact, if an analogous inequality could be proved for area- minimizing surfaces of higher dimension and codimension, then one could just as easily prove the uniqueness of tangent cones and other strong conditions on the local combinatorial and differential structure of the surfaces. So it is a Reifenberg-type inequality, comparing the area of a surface within a ball to that of the cone over its boundary, that underlies the solution to the problem, and it is the translation of the problem from one of cones in balls to one of simpler cones having nice projections that constitutes the major progress involved in proving the result outlined in this paper. ACKNOWLEDGEMENT This work was supported in part by a National Science Foundation Graduate Fellowship. REFERE NCE S [ 1] PLATEAU, J. A .F ., Statique Expérimentale et Théorique des Liquides Soumis aux Seules Forces Moléculaires, Gauthier-Villars, Paris (1873). [2] ALMGREN, F.J., Jr., Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints (in preparation). 13] TAYLOR, Jean, Regularity of the singular set of two-dimensional area-minimizing flat chains modulo 3 in R3, Ph.D. thesis, Princeton University (1973). [4] TAYLOR, Jean, The structure of singularities in soap-bubble- and soap-film-type minimal surfaces (in preparation). [5] REIFENBERG, E.R., An Epiperimetric Inequality related to the analyticity of minimal surfaces, Ann. Math. 80 (1964) 1. [6] REIFENBERG, E .R ., On the analyticity of m inim al surfaces, Ann. Math. 80^(1964) 15. [7] FEDERER, H., Geometric Measure Theory, Springer-Verlag, New York (1969). IAEA- SMR-11/35 CAUSTICS J. GUCKENHEIMER Institute of Mathematics, University of Warwick, Coventry, Warks, United Kingdom Abstract CAUSTICS. After a description of light caustics as they arise in geometric optics their relationship with Riemannian geometry is described; the Hamilton*Jacobi theory is dealt with and the solutions of first-order partial differential equations are discussed. Finally, the connection of catastrophes and caustics is described. This is an expository paper on the singularities of solutions of first- order (non-linear) partial, differential equations. The first section of the paper is a description of light caustics as they arise in geometric optics. The relationship of light caustics with Riemannian geometry is described. The second section deals with the Hamilton-Jacobi theory and the solutions of first-order partial differential equations. The final section explores the relationship of singularities of solutions of first order PDE's with Thom's theory of catastrophes. We do not document the history of these ideas, but only remark that their origins are deeply embedded in optics, mechanics, and differential geometry. The conceptual innovation of the past few years which allows the subject to be treated smoothly is the concept of a Lagrangean manifold. This idea seems first to appear in the work of the Russian physicist Maslov and has been studied subsequently by Arnold [1], Weinstein [14, 15], and Hormander [5]. A brief introduction to Thom's theory of catastrophes appears in Ref. [11]. A more detailed version of our work in this direction will appear in Ref. [4] 1. LIGHT CAUSTICS Geometric optics is a method for studying approximate solutions of Maxwell's equations. To emphasize the geometric nature of the theory, we describe it in the co-ordinate-free context of a Riemannian manifold M. (The index of refractivity of M determines its Riemannian metric.) In the theory of geometric optics, light consists of particles which propagate along light rays. The paths of light rays are geodesics. They can be described from the integral curves of a Hamiltonian vector field defined on the cotangent bundle T*M of M. Recall the procedure: The metric on M induces an isomorphism of the tangent and cotangent bundles of M. Using this isomorphism, we regard the metric of M as defining an inner product 281 282 GUCKENHEIMER ( , ) on each fibre of T*M. Define the function H: T*M -* R by H (u) = I ш У . Then H is a smooth function which we take as the Hamiltonian function for defining a Hamiltonian vector field XH on T*M. The Hamiltonian vector field XH is defined as follows: There is a canonical two form S7 on T*M. (See section 2 for the definition of Í2.) П is closed and of maximal rank; hence it defines a symplectic structure on T*M. Define the isomorphism £ ; T (T*M) - T*(T*M) by £(Xi) (X2) = Q(XU X2) for all Xi, X 2 G Tp (T*M), p€T*M. The map I is an isomorphism since Q is a bilinear form of maximal rank on each fibre of the tangent bundle of T*M. We now define XH = JT1 (dH). The geodesics of M are the projections of integral curves of XH. The integral curves of XH lie on the level surfaces of H in T*M. Restricting our attention to a specific level surface (say H = 1) corresponds to fixing the speed of light. The usual description of the geodesic flow of M takes place in the tangent bundle rather than the cotangent bundle. By working in the cotangent bundle we can reinterpret the meaning of the geodesic flow in terms of wavefronts and Huyghens' principle. Up to a scalar factor, each non-zero covector at pGM is determined by its kernel which is a hyperplane in the tangent space to M at p. Now think of a wavefront of light Wpropagating through M. W is a hypersurface of M and its tangent space at each point is a hyperplane in TM. There is a line of covectors in T*M at each point of W having the tangent space of W as kernel. Choosing the direction of propagation of W and the speed of light gives us a unique covector associated to W at each point. This covector is defined as one of the points of intersection of the line of covectors annihilating the tangent space of W with the sphere whose radius is the speed of light. Thus there is a map j: W -» T*M. If we follow j(W) along the geodesic flow for time t and project this submanifold of T#M into M, we obtain the new position Wt of the wavefront W after it has moved for t units of time. The isomorphism between T*M and TM induced by the Riemannian metric associates to each non-zero covector a normal to its kernel (of the same length and with the proper orientation). In terms of a propagating wavefront of light this implies a form of Huyghens1 principle: a wavefront of light propagates normal to itself. For a treatment of catastrophe theory from this point of view, see Ref. [9 ]. It is possible to solve for the successive wavefronts Wt CM more directly than we did above by introducing the Hamilton-Jacobi partial differential equation. In our case of geometric optics, we seek a smooth function f : M -* R such that \ To illustrate the nature of the problem, consider an example. Take the parabola P = {(x, y ) | y = x2} as initial wavefront W in the plane M with the direction of propagation being upwards. To calculate the function f satisfying the partial differential equation 3f о 9f 2 f— Г + (— ) 1 Эх 1Эу; with f j P =0, consider the function g :R -* R defined by g(x, y, s) = [(x-s )2 + (y - s 2)2 The variable s parametrizes the points of P, and g determines the distance from the point (s, s2) on P to the point (x, y). At a point (x, y) on the wavefront Wt at time t, a segment of length t from (x, y) to a point (s, s2) will be a ray if the segment is perpendicular to P at (s, s ). T his m eans that 9 S ,(x, y, s) =0 9s for such a segment. It follows that f is obtained from by solving this last equation for s and substituting: 1 - 2y £ £ = _ * + s + s 9s 2 2 There are values for (x, y) in a neighbourhood of which we cannot solve the equation 3g/9s = 0 smoothly for s. These represent points at which Wt is tangent to a light ray determining Wt. These points are called caustics. In a neighbourhood of the caustics, it is impossible to find a smooth solution of the Hamilton-Jacobi equation ^ d f, dO = 1 with the given initial data. In Riemannian geometry, one can interpret the conjugate locus of a point p as the caustic set of a point source of light at p. Section 3 deals with the question of determining the "generic" local geometric structure of caustic sets. Before doing so, we broaden the context to which the analysis of section 3 will apply. 2. HAMILT ON - J AC OBI EQUATIONS The same techniques used by geometric optics to give approximate solutions to Maxwell's equations can be used to give approximate solutions to more general partial differential equations [5, 6 ]. The approximate solutions are found by solving a Hamilton-Jacobi equation for the characteristics of the partial differential equation. With this brief motivation, we proceed to summarize the Hamilton-Jacobi theory. We are interested in equations which can be written locally in the form H(x, ux) = 0 with x = (xb ..., xn)R n, u:R n ->R, and ux = (9u/9xi, ..., 9u/9xn). Given H :R 2n -» R, a function u :Rn -*■ R such that H(x, ux) = 0 is called a solution 284 GUCKENHEIMER of the Hamilton-Jacobi equation with the Hamiltonian H. Let us now write such an equation in a co-ordinate-free manner on an n-dimensional m anifold M. We in terp ret H a s a function on the cotangent bundle T* T(T*M) The m aps 7Г2, and 7Г3 are bundle projections; лц is dw2. Define the canonical one form и on T *M by u(X) = (/гз(Х))(7г4(Х)) The canonical two form Г2 on T*M is defined to be du. Í2 is a closed two form of maximal rank. If we choose co-ordinates {qi, . . ., qn} on M and then take linear co-ordinates {pi, . . ., pn} in each fibre of T*M relative to the basis {dqi, . . ., dqn}, then n U = Z/ pidqi i=l where dq¿ is now interpreted as a one form on T#M instead of M. In these local co-ordinates. Cl = Zy dpi A dq¡ The reason for introducing Cl at this point is the following lemma: Lemma: Let в be a one form on M, and let i:graph (в) -» T*M be the inclusion. Then в is closed if and only if i*f2 s 0. Proof: We verify the lemma in the local co-ordinates introduced above. In these co-ordinates write в = Ea¡ dq¡ for some a¡ : M -» R. (Note that dq¡ is a one form on M in this expression.) The graph of в is the set of point with co-ordinates (qi, ..., qn, ai(q), ..., an(q)). The tangent space to graph (в) is spanned by Г Э ул 9aj Э 1. 1 9q¡ j 9q¿ 9pj i To evaluate we must compute £2(Х ь X 2) for Xi, X 2 tangent to graph (0). Now, Q ( ± . +y ^ + У ) Эа^ 9a¿ V 9qt 4-, 9qt 9р; ' 9qk 9qk 9p¡ J 9Qí 3qk IAEA-SMR-11/35 285 The form в is closed if and only if Эа^ Э аj 9qi 9qi( This is satisfied if and only if i*f2 = 0, as is evident from the above formula. Poincare Lemma: A closed one form в on M can be written locally a s du for som e function u : M -> R. Consequently, we obtain the following characterization of solutions of H = 0: an n-dimensional submanifold i : X -* T*M locally represents a solution of H = 0 if 1) H I X = 0 2) i* Q = 0 3) X is transverse to the fibres of T*M. Condition 3) guarantees that X is locally the graph of a one form on M; 2) is the condition for that one form to be locally exact. We propose now to generalize the definition of a solution of H = 0 by simply forgetting condition 3). Definition: A Lagrangean submanifold of T*M is an n-dimensional submanifold i :X -* T*M such that i*f2 = 0. A solution of H = 0 is a Lagrangean submanifold of T*M such that H° i = 0. The singular set of Xis the set of points at which X is not transverse to the fibres of T*M. The singular set consists of those points of X at which X cannot be written locally as graph(du), u:M-*-R. In analogy with the definition of a light caustic, we define the caustic set of X to be the projection of the singular set of X into M. In the next section, we shall consider generic caustics of solutions. To speak of genericity, it is necessary to give the set of solutions the structure of a topological space. Within the framework we have developed, this can be done in a standard way. The smooth embeddings of an n-dimensional manifold N in T*M form an infinite dimensional manifold, and those embeddings which are Lagrangean form a closed submanifold of this infinite dimensional manifold. If one wishes to work with unpara metrized submanifolds, we may divide by the free action of the diffeo morphism group of N. The set of solutions of H = 0 which are diffeo morphic to N inherits a topology from this space. In this topology, two submanifolds Xb X2 are close if there is a diffeomorphism i: Xi -» X2 which is C“ -close to the inclusion i: Xi ->■ T*M in C°°(Xi, T*M). Before proceeding to the discussion of caustics in the next section, let us remark that there is a slightly different problem which arises in Hormander [5] and can be dealt with in an entirely analogous manner. In this problem, one obtains a Hamiltonian H which is a positively homo geneous function of the fibre co-ordinates in T*M. This means that if f GTq*M and с > 0, then H(?) = H(cf). One then looks for positively homogeneous Lagrangean manifolds which, are solutions of H = 0. A Lagrangean manifold is positively homogeneous if it is a union of rays in the fibres of T*M. Such a manifold cannot be transverse to the fibres of T*M; so its caustic set is the entire image of its projection into M. The 286 GUCKENHEIMER set of positively homogeneous Lagrangean manifolds again has the structure of an infinite-dimensional manifold; hence, genericity makes sense in this context. The results stated in the next section apply to both cases. 3. CATASTROPHES AND CAUSTICS Our aim is to describe the local structure of generic caustics of solutions of a Hamilton-Jacobi equation. The main result is the following, where "generic" means belonging to some open-dense set of the appropriate topological space. Theorem: For a generic Hamiltonian H, the caustic set of a generic solution of H = 0 has the local structure of the product of a disk with an elementary c-catastrophe of Thom. We now recount enough of Thom's catastrophe theory to define an elementary catastrophe. Let f : Rn-> R be a function having a critical point at 0. If 0 is a non-degenerate critical point, then any perturbation of f will have a unique, non-degenerate critical point near 0. However, if 0 is a degenerate critical point of f, then a perturbation of f may have several critical points near 0 or none at all. For example, if f: R -» R is defined by f(x) = x5, then a perturbation of f may have 0, 1, 2, 3, or 4 critical points near 0. Thom's catastrophe theory allows us to analyse the behaviour of (degenerate) critical points under perturbation. Consider the space P of proper C“ functions f: Rk -» R. The groups Dk and Di of C” diffeomorphisms of Rk and R act on P by composition on the right and left, respectively. That two points of P lie in the same orbit of the action of X Dj means that the two functions differ only by co-ordinate changes. The orbits of D^ X Di give P a stratified structure. The open orbits are called stable functions. It is a theorem of Morse theory that the stable functions are those with non-degenerate critical points and distinct critical values. We define the codimension of fGP to be the codimension of its orbit under the action of D^XDj. Let f€P have codimension m. Then we choose a map Ф : Rm -► P such that$(0) = f and such that Ф is transverse to the orbit of f at f. The map Ф is called a universal unfolding of f. The inter section of the image of Ф with the various orbits of Dk X Di defines a stratification of Rm. In particular, consider the set ССФ(ЙШ) of non-stable functions. С is a stratified set with a natural stratification (which may be different from that given by its intersection with the orbits of D^ X Di. ). Choose RncRm such that Rn is transverse to the stratum of 0. Then CDRn is called the elementary catastrophe of f. Its germ at 0 is determined up to isomorphism of stratified sets. Usually, one studies the local problem which arises from choosing f to have a single degenerate critical point and distinct critical values. Mather has proved that, for m S 6, the stratum of 0GC consists of 0 itself and, consequently, m = n. It follows from Mather's genericity theorem that, up to isomorphism of stratified sets, there are only a finite number of elementary catastrophes of a given dimension. Thom has given names to the seven elementary catastrophes of dimensions at most four. As we have indicated above, Morse theory gives two conditions which are necessary and sufficient for the stability of a function f: (1) f has non-degenerate critical points only, and ( 2) no two critical points lie on the IAEA-SMR-11/35 287 same level surface of f. The catastrophe set of a non-stable function f splits into two (non-disjoint) pieces corresponding to functions which have degenerate critical points and to functions which have two critical points with the same value. The first piece we call the c-catastrophe set of f; the second Thom calls the Maxwell set of f. It is the c-catastrophe set which appears in our theorem. These are the bare abstract outlines of Thom's theory. Practically, one can be much more specific in giving "normal" forms for functions and their unfoldings having a given catastrophe. Briefly, this is done as follows. One can take for f a polynomial defined on Rk (k depends on the catastrophe) such that f(0) = df(0) = d2f(0) = 0. Let I be the ideal generated by the first partial derivatives of f in P. In order for f to have finite codimension, it is necessary that the dimension of P/I be finite. Choose a set of polynomials v0, . . . , vm which project onto a basis of P/I with v0 a constant function. Then the unfolding of f is given by m Ф (аь . . . , am) = f + £ a iVi i=l Since d2f = 0, we may take Vj as the i-th co-ordinate function of Rk for 1 á i s k. Later, we shall choose a different normal form more suitable for application to the Hamilton-Jacobi theory. Let us look more closely at a family of functions Ф : Rm -* P. Corres ponding to Ф is a map F : Rk X Rm R X Rm defined by F(x, t) = (Ф(t)(x), t). Denote Tri°F by Fi where n\ : R X Rk -* R is the projectio n . The se t E of critical points of F is the set of points (x, t) for which - ° This is the union of the sets of critical points for the functions Fi (•, t) :Rk -*• R. Define the map a : E-* T*Rm by tf(x,t) = (t, |fi(x,t)) Proposition: 1) For generic families of functions Ф :Rm -*■ P , a defines a Lagrangean submanifold of T*Rm . 2) If к Sm, the corresponding map from germs of families of functions to germs of Lagrangean submanifolds of T*Rmis surjective. We omit the proof of this proposition (which is not difficult). This proposition gives us the means of using the catatrophe theory to study the caustics of Lagrangean manifolds. Locally, every Lagrangean manifold arises from the set of critical points of a family of functions. To study the caustic set, one has the following: Proposition: Let Ф be a family of functions so that Е(Ф) is a manifold. Then the singular points of ct(E) (as defined in section 2) are the images of points (x, t) for which F(-, t) has a degenerate critical point as x. Corollary: The caustic set of cr(E) is isomorphic to the c-catastrophe se t of Ф . 288 GUCKENHEIMER The proposition and corollary imply that the study of caustics of generi Lagrangean manifolds is locally equivalent to the study of the c-catastrophe sets of generic families of functions. The only remaining tool needed for the proof of the theorem is the Thom transversality theorem. The trans versality theorem implies that, for generic Hamiltonians, the generic solution of each of these is a generic Langrangean manifold. Using the normal forms for unfoldings of singularities, one can write normal forms for germs of generic Lagrangean manifolds of T*Rn. If XcT*M is a Lagrangean manifold through p, then one may choose canonical co-ordinates so tfiat X is transverse to the constant section of T*Rnat p. This means that locally X can be written as graph (dh), where now h :Rn* -» R and we identify (Rn*)*with Rn. Consider the family H (x, Ç) = h(?) - Z) Xi?i for xGRn and |G R n*. We then have that graph (dh) = | | | , Ç J which is also the set E(H) of those (x, §) for which Thus, in the second part of the first proposition above, we may always choose the family of functions to be of the special form of H in which the unfolding parameters appear as linear combinations of the co-ordinate functions in the domain. We can obtain normal forms of this type for generic Lagrangean manifolds corresponding to families of functions for which the stratum of 0 in the unfolding space was an isolated point. To do this, take the normal form n î - Z / » iV i i=l which we obtained before for f :Rk ->■ R with v4 = = i-th co-ordinate of Rk for 1 s i s k. If к = n, this is of the desired form. Otherwise, if к < n, replace f by g : Rn -* R defined by П g(xb . . . , Xn) = f(Xl, . . . , Xh) - \ Z / ( X j - V j ( x b . . . , Xk))2. j=k+l Then g has the same catastrophe set as f and the unfolding of g is given by g - f j а Л i=l As a final remark, we note that the theory we have described allows one to place Thom's theory of catastrophes into a dynamical framework without resorting to his "static model" [11]. This approach avoids some of the technical difficulties which seem to be related to the static model [3] IAEA-SMR-11/35 289 ACKNOWLEDGE ME NT The author would like to thank the University of Warwick for kind hospitality extended during preparation of this paper. REFERENCES [1] ARNOLD, V .I., Characteristic class entering in quantization conditions, Functional Anal. Appl. 1 (1967) 1. [2] DARBOUX, Mémoire sur les solutions singulières des équations aux dérivées partielles du premier ordre, Mémoires de l'Institut Sav. Etrangers (1883). [3] GUCKENHEEMER, J. ."Bifurcation and catastrophe", Dynamical Systems (PEIXOTO, M.M., Ed.)(1973 ) 95. [4] GUCKENHEIMER, J ., Catastrophes and partial differential equations, Ann. Inst. Fourier 23 (1973) 31. [5] HORMANDER, L ., Fourier integral operators I (especially section 3 .1 ). Acta Math. 127 (1971) 79. [6] HORMANDER, L ., DUISTERMAAT, Fourier integral operators II (to appear). [7] LATOUR, F ., Stabilité des champs d’ applications différentiables; généralisation d'un théorème de J. Mather. C.R. Acad. Sci. Paris 268 (1969) 1331. [8] MATHER, J ., Stability of mappings I-VI. I: Annals of M athematics 87 (1968) 89. II: Annals of M athem atics 89 (1969) 254. Ill: Publ. M ath. IHES 35 (1968) 127. IV: Publ. Math. IHES 37 (1969) 223. V: Advances in M athem atics 4 (1970) 301. VI: Proceedings of Liverpool Singularities Symposium, Springer Lecture Notes in Math. 192 207. [9] PORTEOUS, I ., Normal singularities of submanifolds. J. Diff. Geom. J> (1971) 543. [10] THOM, R., Stabilité Structurelle et Morphogenèse, Addison-Wesley. [11] THOM, R., Topological models in biology. Topology 8 (1969) 313. [12] THOM, R ., LEVINE, H ., Lecture notes on singularities, Proceedings of Liverpool Singularities Symposium, Springer Lecture Notes in Math. 192. [13] WALL, C .T .C ., Lectures on С-stability and classification. Proceedings of Liverpool Singularities Symposium, Springer Lecture Notes in Math. 192 178. [14] WEINSTEIN, A ., Singularities of families of functions. Berichte aus dem Mathematischen Forschungs- institut 4 (1971) 323. [15] WEINSTEIN, A ., Lagrangean manifolds. Adv. M ath. 6 (1971) 329. IAEA-SMR-11/36 THE TOPOLOGICAL DEGREE ON BANACH MANIFOLDS C.A.S. ISNARD Instituto de Matemática Pura e Aplicadaj Rio de Janeiro, Brazil Abstract THE TOPOLOGICAL DEGREE ON BANACH MANIFOLDS. The degree of a proper C^Fredholm map is defined and its invariance, homotopy and multiplicative properties are proved. Some transversality theorems of independent interest are proved. The Sard-Smale theorem for proper maps follows as a corollary. INTRODUCTION The subject of this article is a definition of a degree for proper C^Fredholm maps between C1-Banach manifolds, extending the work of Smale [33], Browder [3], Elworthy [13] and Tromba [13]. The results obtained here are the base for the construction, in Refs [42, 44], of a degree for proper continuous maps of the type f+c, from C 1-Banach-manifolds to subsets of Banach spaces, where f is a C-^-Fredholm map and с is a locally compact map (perhaps non-differentiable). That degree coincides numerically with the Leray-Schauder degree, [22], which is the particular case where f is the identity on some open subset of a Banach space, and с is globally compact. Smale's, Elworthy's and Tromba's degrees required the maps and manifolds to be of class C2 . Browder's degree was defined for C 1-Fredholm- map of index 0, but it required the domain to be an open convex subset of a Banach space. It was his suggestion of the non-convex case as a doctoral dissertation topic that started this research. The class of maps between oriented C1-manifolds for which one can define an integer-valued degree is expanded here to the "C 1-® *-m ap s" (further extensions can be found in Ref. [42]). The main tools in the constructions of the degree are the transversality techniques in sections D and E, which reduce the problem to finite dimensions, i.e. to Brouer's degree. The same techniques are used in Refs [42, 43] to reduce from finite dimensions to dimension 1 , where degree theory is a trivial consequence of order. Through the reduction to finite dimensions we also re-obtain here Sard-Smale's theorem for proper Fredholm maps from Sard's result [31]. The setting of manifolds, rather than Banach spaces, is a requirement of the methods employed, because the pre-image of a linear subspace by a non-linear transversal C1-map will be a C 1-manifold, perhaps non-linear. The degree of complex differentiable Fredholm maps is also included, giving again some of the results in Ref. [ 13]. As was pointed out in Ref. [ 33], the degree of C k+1 Fredholm maps of index к > 0, at regular values, is an element of the non-oriented cobordism additive group r)k of k-dimensional compact С ^manifolds defined by Thom in 291 292 ISNARD Ref. [ 39]. When the С k+1-manifolds are oriented and the maps are the С к+1 -Ф*(Т)-тарв, where T is surjective and has a k-dimensional kernel, we likewise define an oriented degree, taking values in the corresponding oriented cobordism group f2k. We have an oriented degree also in the complex case, through the interpretation of each k-dimensional complex manifold as a 2k-dimensional real oriented manifold. In the text the statements of the degree theorems are all for the index-0 case, but we state and prove all the auxiliary theorems in the more general context, so that, by a simple replacement, one obtains the degree theories of maps with positive index. A. THE SETTING FOR THE DEGREE X and Y are arbitrary topological spaces, and X and Y are open subsets of X and Y, respectively, that in the topology from X and Y, respectively, are real, real-oriented, or complex C1-Banach manifolds (definitions in sections A and B), with X also supposed to be a Hausdorff space (X-X and Y-Y may correspond, e. g. to the boundaries of the manifolds, or may be empty). When G is any open subset of X, G is the closure of G in X and 9G = G - G. — A A We call a map f: X -*■ Y proper if for every compact subset К of Y the se t f _1(K) is compact. The degree will be defined for continuous proper maps f: X -* Y that have for restrictions C1-$0-maps X -» Y. Section A will be spent in definitions, of Ck-Banach-manifolds (k S 1, integer or +°o), and Ck-®m-maps (m integer). The topologies on Banach spaces and on their subsets will always be the metric ones given by the norms. If Eo and E are real or complex Banach spaces, Ь(Ец, E) is the Banach space of all continuous linear operators T: Eo -*■ E, with norm defined by | T | = sup{ | Tu | | | u | = 1}. A Fredholm operator is any T e L(E 0, E) such that dim N(T) < +oo and dim(E/R(t)) < +00, where N(T) and R(T), respectively, denote the null-space and the range of T. The index of T is dim(N(t)) - dim (E/R(T)), and «ÊmiEo, E) is the set of the Fredholm operators E 0 -* E of index m. If Eo or E are finite-dimensional and Фт(Е0, E) f p, then dim E 0 = m + dim E and L ( E 0, E) = Фт (Е0,Е). If X is open in Eo and f: X -* E is a Ck-map such that f'(x) is in some subset a of L(E0, E) for all x in X, then f: X -» E is called a Ck-q-map. The С^Ф^Ео, E)-maps are also called Ск-Фт-тар 8. A point x e X is called a regular point for f: X -> E if and only R(f' (x)) = E. If f is а С 1-Фо-тар, then x is a regular point for f if and only if f'(x) is a one-to-one operator, and if and only if f'(x) is an isomorphism onto E, because if T e Ф0(Е0,Е), thenN(T) = 0«-> R(T) = E*-> T is an isomorphism E 0 = E *- R(T) = E. A real or, respectively, complex atlas of class Ckfor some topological space X is any collection of homeomorphisms a : Ua = а(и а ), called the charts of the atlas, where 1) The Ua form an open cover of X, and each o(Ua) is an open subset of some real or complex, respectively, Banach space Ea; 2) If n Ua2 f p for any charts а-у, ot Y is а Ск-Фт-тар if and only if dim X = m + dim Y. A regular point for a Ck-map f: X -» Y (k s 1) is any x e X such that a(x) is a regular point for some composition (3 • f ■ а"г of f with charts at x. B. THE SETTING FOR THE ORIENTED DEGREE A degree taking value in Z 2 (= integers modulo 2) will be defined for all the maps satisfying the general conditions of section A. Such degree shall be called a mod 2 degree. Sometimes there is an integer-valued degree that, taken modulo 2, gives the corresponding mod 2 degree. One such case is when the manifolds are complex. The other case is the subject of this section: when X and Y are oriented C1-manifolds and the map f: X -> Y satisfies some condition that permits us to assign one positive or negative orientation to each of its regular points (in a locally constant way, coherent with finite-dimensional transversality). We call this the oriented case, and the corresponding integer-valued degree is the oriented degree. When X and Y are oriented C1-manifolds modelled after Rmand f: X -* Y is a C^map, the orientation of any regular point x e X is the sign of the determinant of the Jacobian at a(x) of an arbitrary (3 f a"1, composition of f with charts at x. In infinite dimensions the situation complicates because the determinants are not always available. Definition: If E and E! are Banach spaces, Lc (E, Ei) is the set of compact operators E -» E 1( which are the T e L(E, Ej) such that C({x e E | | x| S i}) is contained in some compact subset of E i. We call Lf(E, Ei) the set of all 294 ISNARD T 6 L(E,Ei) with finite rank (rank (T) = dimR(T)). Then Lf(E, Ei) £LLC(E, Ei), and those two sets are linear subspaces of L(E, Ej). For any T e L(E, E x) we call T + Lf(E, Ej) or T + LC(E, Ej), respectively, the subset {T + С I С eLf(E, Ei) or L C(E, Ei) respectively} of L(E, Ei). In Ref. [ 20] it is shown that L C(E, Ej) is closed in L(E, Ej), that if T e Фт(Е, Ej), then T + LC(E, Ej) с Фт(Е, Ej), If T jeLfE .E i) and T2 eL(Eb E2), where E, Ex and E 2 are Banach spaces, then T2 Ti is compact or, respectively, has finite rank if either T 2 or Ti is compact or, respectively, has finite rank. Definition: If T e L(F, F) where F / 0 is finite-dimensional, we define det(T), the determinant of T, to be the determinant of any matrix representation of T obtained by choosing a basis for T. The value of det(T) is independent of the chosen basis I: E -* E will always denote the identity operator. Definition: If T e I + Lf(E, E), where E f 0 is any Banach space, let F / 0 be any finite-dimensional linear subspace of E containing the range of T-I. Then T (F)£F and we define det(T), the determinant of T, to be the deter minant of the restriction F -> F of T. To show that det(T) is iridependent of the choice of F, suppose F 0 = R(I-T) / 0. Since FqÇF, it follows that it suffices to prove the statement when E itself is finite-dimensional. Then E = F x E 0 for some Eq, so it suffices to prove that if F and Eoare finite dimensional, and T 6 L(F x E 0, F x E 0) is of the type T(u, v) = (Tu + Sv, v) for some T e L(F, F) and S e L(E0, F), then det T = det T. This follows immediately from properties of determinants. Proposition: If T j, T2 £ I + Lf (E, E) then T jT 2 £ I + L f (E, E) and det(T 1 T 2 ) = det (Tx ) • det (T2). Proof: Take F / 0 containing R(Tj-I) + R(T 2-I), F a finite-dimensional linear subspace of E, then R(T 1 T 2 -I) £F, and consider the restrictions F -*F of the operators. Definitions: GL(E) = (T e L(E, E) | T : E = E is isomorphism}, GLf(E) = GL(E) n (I + L f(E, E)), GLC(E) = GL(E) n (I + L C(E, E)). Proposition: If E / 0 then GLf(E) = { T e I + L f(E, E) | det T f 0 }. Proof: Suppose T 6 I + Lf(E, E), then R(T-I) £F f 0, F finite-dimensional £E . Call T : F -* F the restriction of T, then det(T) = det(T), so it suffices to prove that T 6 GL(E)*-*T £GL(F), Since dim F < +00, we get from Ref. [ 20] that E = E0 ® F for some E 0. So E = E0 x F, hence it suffices to prove: Lemma 1; Suppose F x, F2, E 0 are Banach spaces and T 6 L(Fj xE0, F 2 x E 0) is of the type T( u, v) = (Tu + Sv, v) for all u sF j and veE o, for some T : Fi -* F 2 and S : E 0 -* F2. Then T is isomorphism Fj x Ец = F 2 x Eo Ï is isomorphism Fj = F 2 . Also T e ®p(Fj x E9, Fj x E0) ” T e *p(Fi, F 2). Proof: T = Ho (T x I), whereJT x I ê (Fj x E 0, F 2 x E 0) and H e L (F 2 x E 0, F 2 x E 0) are defined by (Txl)(u, v) = (Tu, v) and H(w, v) = (w + Sv, v). Claim that H e G L(F2 x E 0). In fact, H = I + A, where A: F 2 x E ( - * F 2 x E 0 is defined by A(w, v) = (Sv, 0), and from A о A = 0 we get (I + А) о (I - A) = I = (I - А) о (I + A). IAEA-SMR- 11/36 295 So T: F 2 x E 0 = F 2 x E 0 is iso m o rp h ism *-* T x I: F 1 xE o = F 2 x Eo is is o morphism «-* T: Fj = F 2 is isomorphism. Also, dim N(T) = dim N(TxI) dim N (T), F 2 x Eo F 2 x Eo _ F 2 dim R(T) - dim r (îji xI) - dlm R(^) Proposition: When E f 0 is a real Banach space then GLf(E) = GLf(E) и GLj(E), where: Definition: GLf(E) or, respectively, GL}(E) = { T e i + Lf(E, E) | det(T) > 0, det(T) < 0}. To define an oriented degree, we need to have orientations assigned to the regular points of the map. Those orientations are given by the derivatives of the compositions with charts, so those derivatives must be in some subset a of L(E, E) such that ст П GL(E) has two path-components, one containing the identity I, the elements of which are called orientation-preserving, the other component consisting of the operators that are called orientation-reversing. We cannot take a = Ф0(Е, E) or ст = L(E, E), because in those cases a nGL(E) = GL(E), and GL(E) has only one path component when E is any infinite-dimensional Hilbert space [19], or is üp or Co [ 26]. It is well-known [37], [42] that GLf(E) and GLC(E) have two path- components each, which are GLf(E) and GLf(E), and resp. GLc(E) and GL'C(E), those last two defined by GLf(E) Q GL*(E) and GLj(E) çgl'JE ). We introduce a larger set with two path-components, Ф*(1) n GL(E), an open subset of L(E, E), where Ф*(1) will be defined next. If Eo and E are Banach spaces and T: Eo -* E is a Fredholm operator we define Ф^Т), the Fredholm star of T to be the set of all S such that tS + (l-t)T is a Fredholm operator Eo-» E when 0 S t S 1. Ф*(Т) is then a star-like set of Fredholm operators. If S e Ф*(Т), then T + Lc(Eo, E )çï,(T ). Ф„,(Т) is an open subset of L(E, Ex), because so are all the sets Фт (Ео, E), by section E, lemma 1. For any real Banach space E / 0, each set in the inclusion GLf(E) ç GLC(E) С Ф*(1) n GL(E) has exactly two path-components (Appendix). (Actually, in Ref. [42], it is shown that those inclusions are homotopy equivalences, so that, when E is any infinite-dimensional real Banach space, all those sets have the homotopy type of GL(°o), with homotopy groups given by the Bott periodicity theorem. ) The sets of orientation preserving or, respectively, orientation reversing isomorphisms, that are, by definition, the path-components of Ф*(1) n GL(E), are open in L(E, E). When E f 0 is a finite-dimensional real Banach space every isomorphism gets an orientation characterized by the sign of the determinant, because I + Lf (E, E) = L(E, E) = Ф0(Е, E) = Ф„(1) and GLf(E) = GL(E). Definition: А С к -Ф^-тар is а С к -Ф„(1)-тар. The oriented degree will be defined for proper С1-Ф *-тарз between oriented C1-manifolds modelled after the same real Banach space E (oriented manifolds are defined in section A). Further generalizations can be found in Ref. [42]. 296 ISNARD Definition: Let f: X -► Y be а С^-Ф^-тар and X and Y be oriented C1-manifolds modelled after the real Banach space E f 0. We say that xe X is a regular point for f with positive or, respectively, negative orientation when there are ch arts a and /3 in atlases for X or, respectively, Y, with x in Ua and f(x) in Ue, such that )3fa 1 has for derivative in a(x) an orientation preserving or, respectively, reversing isomorphism E = E. That each regular point has a uniquely defined orientation, independent of the chosen charts, follows from: Multiplication law of orientations: Let E / 0 be any real Banach space, T e Ф*(1) n G L(E), and U 6 GLC(E). Then TU and UT are orientation-preserving if T and U have the same orientation, and TU and UT are orientation-reversing if T and U have different orientations. Note 1: This law does not hold for unrestricted T and U in Ф^Ш. In fact, it is easy to show that when GL(E) is path-connected, as it is the case with ip, with Co, and with all infinite-dimensional Hilbert spaces, then any isomorphism E = E is equal to the composition of finitely many orientation- preserving isomorphisms in Ф*(1). Proof: TU and UT e Ф*(1), because if C e LC(E, E) then TC and CT e LC(E, E) (from Ref. [20]). There are continuous paths t -» Tt in GL(E) n Ф^(1) and t -* Ut in GLC(E), connecting T, (or U, respectively) to elements of GLf(E). Then t -* T t Ut and t -» Ut Tt are continuous paths in GL(E) n Ф„.(1). Since orientations are constant along those paths, it suffices to prove the multi plication law when T and U are both in GLf(E). In this case, it is a consequence of det(TU) = det(T) . det(U) = det(UT). The orientation-preserving Ck-maps are the Ск-Ф,,.-тарз (ks 1 ) for which every regular point has positive orientation. In case X and Y are open subsets of Rn, this means that det(f'(x)) ê 0 at all x e X. An orientation-preserving С k-diffeomorphism is any orientation- preserving Ck-map that is also Ck-diffeomorphism. The inverse map is also an orientation-preserving Ck-diffeomorphism, because if T is in Ф*(1) n G L(E), so is T " 1, and then T and T "1 have the same orientations: In fact, if 0 И И then T ' 1 (tT + (l-t)I) = tl+ (l-t)T" 1 e$o(E,E), hence T 1e Ф*(1), and the m ap T *-* T "1 is a homeomorphism GL(E) п Ф„.(1) = = GL(E) n Ф*(1), so it makes path components correspond to path components. Re mark: It is easy to show that Ф*(1) contains I+H whenever He L(E, E) has essential spectral radius Pe(H) < 1 (pe(H)) is the spectral radius of H + LC(E, E) in the Banach algebra L(E, E )/L C(E, E), i. e. pe(H) = lim 1 Hn|c , where IT |c = |Т+ L c(E, E) I = inf { IT + С I I C e LC(E, E)}. It is always true that Pe(H) S |h | c - |Н |). C. DEFINITIONS OF THE DEGREE AND APPLICATIONS In all the definitions of the degree in this section, X and Y are open subsets of X and Y, respectively, which are arbitrary topological spaces. The topologies on X and Y are always those which they receive from X and Y, respectively, and the topology on X is always supposed to be of the Hausdorff type. We have a continuous proper map f: X -♦ Y and we suppose that one of the following three conditions is satisfied: IAEA-SMR-11/36 297 (i) The general case: X and Y are arbitrary C1-Banach-manifolds and the restrictions X -*• Y of f is а С 1-Фо-тар. The degree is an integer m odulo 2. (ii) The complex case: X and Y are complex C1-Banach-manifolds and the restriction X -> Y of f is a С^-Фо-тар. The degree is a non-negative in teger. (iii) The oriented case: X and Y are oriented C1-manifolds modelled after the same real Banach space E f 0, and f: X -* Y is а С 1- Ф *- т а р . The degree is an integer. Remark: When X and Y are modelled after the same finite-dimensional space, any C1-map X -* Y is а С1-Фо-тар and is а С1- Ф *- т а р . Definition: A point of X is a critical point for f if it is not a regular point. A critical value for f is the image by f of a critical point. The regular values for f are the elements of Y - f(3X) that are not critical values. So ye Y is a regular value for f if and only if every point of f "1 (y) is in X and is a regular point for f. In particular, y is a regular value for f whenever f "1 (y) is empty. The degree of f on X at y, written deg(f, X, y), will be defined at all y e Y - f(9X). Definition: Suppose ye Y - f(9X) is a regular value for f such that f_1(y) has m elements (m < +°o). Then we define: (i) In the gen eral c ase , deg(f, X, y) = m(m od 2) (ii) In the complex case, deg(f, X, y) = m (iii) In the oriented case, deg(f, X, y) = p-n, where pandn, respectively, is the number of points of f" 1 (y) that a re regu lar for f with positive (negative, respectively) orientation. Theorem 1: If y e Y — f(3X) is a regular point for f then f _1(у) is a finite set, with a number of elements equal, modulo 2, to deg(f, X, y), and у has a neighbourhood consisting of regular values y' e Y — f(3X) for f such that deg(f, X, y') = deg(f, X, y). Proof: We need initially: Inverse mapping theorem: Suppose x is a regular point for а Ск-Ф0-тар or, respectively, а С^-Ф^-тар f: X -* Y (kS 1), where both X and Y are real or complex Ck-Banach-manifolds or oriented Ck-Banach manifolds, respectively. Then x is an some open subset W in X sucn that the restriction of f to W is a С -diffeomorphism onto some open subset f(W) of Y and that every point of W is regular for f, with the same orientation as x, in the oriented case. Proof: Taking compositions with charts, we may suppose that X and Y are open subsets of Banach spaces E and E 1# f'(x) is an isomorphism E = Ej, so x is in some open set X 0 consisting of regular points for f, which all have the same orientation as x, in the oriented case, because the set of all iso morphisms E = E (orientation-preserving and -reversing, respectively) is open in L(E, E), from Ref. [ 20] (or from Section B, respectively). We apply now, to f restricted to XQ, the well-known inverse mapping theorem [20]. 298 ISNARD Proof of theorem 1: Suppose f-1 (y) f 0. Each point of f" 1 (y) is in some open set W in X as in the statement of the inverse mapping theorem, f ' 1 (y) is compact and, therefore, covered by finitely many of those sets W, say W1( ..., Wq, each containing some point of f' 1 (y). Since f is one-to-one on each of those sets W^, it contains exactly one point of f‘ 1 (y). Then f 'Ну) has exactly q elements. We may suppose that the W\ are pairwise disjoint, taking them smaller, if necessary. Call now V = ( (V f(W\)) - f(X - A w *)- Then y£V, and if y'e V, f 1(y') has q points, one on each set Wx. From this we obtain: y' is a regular value for f, y'E Y - f(9X), and deg(f, X, y1) = = deg(f, X, y). We must now show that V is an open subset of Y. We take care of the case f _1(y) = fi showing that Y — f(X) is also open in Y. The proof is then completed by Lemma 1: Let f be any continuous proper map from some topological space to and let Y be an open subset of Ÿ and a Ck-Banach-manifold (кй 0) in the induced topology from Ÿ. Then, for any closed subset A of the domain of f, we have Y — f(A) open in Y and Y n f(A) closed in Y. Proof: Since Y is locally metrizable, it suffices to show that if any sequence f(an) converges in Y to y, with aneA, then y€f(A). The set consisting of y and of the f(an) is compact, so its pre-image by f is a compact set, which contains the an. The an have then a generalized subsequence converging to some x, which must be in A, because A is closed. By continuity, f(x) = y, hence y e f(A). Definition: If y e Y - f(9X) is a critical value for f, define deg(f, X, y) = = lim deg(f, X, yn), where (yn) is an arbitrary sequence of regular values for f, converging to y in Y - f(9X). This limit is well defined, in the sense that deg(f, X, yn) is constant for all n S n0, and this constant value does not depend on the chosen sequence, because of: Theorem 2: Suppose y 6 Y - f(9X). Then y is in some open set V in Y - f(9X) such that deg(f, X, y) is constant for all y in V regular value for f. Sard-Smale Theorem: Every point y e Y - f(9X) is the limit of some sequence of regular values. Those theorems are proved in section D. As consequence we obtain: Invariance property: deg(f, X, y) is constant for y in any fixed connected component of Y - f(9X). Proof: We have deg(f, X, y) constant for any y in the set V given by theorem 2. So the degree is a locally constant function of y, hence it is constant on connected subsets of Y - f(9X). Surjectivity property: If deg(f, X, y) f 0 for some y e Y - f(ЭХ), thenyef(X). Proof: If y ¿'f(X) then y is a regular value. Definition: Let now G be any open subset of X. The restriction G -* Ÿ of f is proper, and has for restriction а С1-Ф0-тар or, respectively, а С 1- Ф *- т а р IAEA-SMR-11/36 299 G -* Y, where G is the real or complex, or, respectively, oriented C 1-B an ach - manifold obtained by restriction of charts. We write deg(f, G, y) for deg(f/G, G, y). We have then a local degree defined. Additivity property: If y e Y - f(X - U Gx), where the Gx form a finite or infinite collection of open subsets of X, then deg(f, X, y) = Ç deg(f, G^, y), and this sum is finite, i. e. deg(f, Gx, y) f 0 for at most finitely many indices X. Proof: f_:L(y) is compact and contained in U Gx, so it is contained in finitely many of the Gx. For the remaining Gx, we have f’ 1 (y) n Gx = 0, hence deg(f, Gx, y) = 0. So we may suppose the collection of the Gx is finite. At a regular value, the theorem follows from the definition of degree. The critical values y are the limit of sequences yn of regular values for f, which we may suppose to be in Y - f(X - U Gx), because this is an open set, by lem m a 1 . Then deg(f, X, yn) = Y j deg(f, Gx, yn). and we take the limit when П -* +00. x Excision property: If A is a closed subset of X such that y^f(A), then deg(f, X, y) = deg(f, X-A, y), whenever y e Y - f(8X). Proof: This is the additivity property with only one Gx, equal to X-A. These properties permit us to generalize a theorem of Cacciopoli [ 5] to the following: Theorem: Let f: X -» Y be a proper С1-Ф()-тар, and suppose that Y is connected and contains a regular point y for f such that f_ 1 (y) has an odd number of elements. Then Y = f(X). The theorem has a stronger form if X and Y are complex C 1-Banach-manifolds, or if f is orientation-preserving, with X and Y oriented. In those cases, if Y is connected and contains some regular value y such that f_1(y) f 0, then Y = f(X). Proof: Since ЭХ = 0 and Y is connected, the invariance property gives deg(f, X, y) is constant for y in Y. For the regular value y in the statement of the theorem, we have deg(f, X, y) f 0. Then, by the surjectivity property, every point of Y is in f(X). Corollary: If Y is connected, X and Y are complex or, respectively, oriented C1-Banach-manifolds and f: X -» Y is a proper С 1-Ф0-тар or, respectively, a proper orientation-preserving C 1-map, then either f(X) = Y or f(X) has no interior points. Proof: By Sard-Smale any interior point will be the limit of some sequence of regular values for f. Some of those regular values will then be in f(X), and we may apply the last theorem to them. D. THE MAIN THEOREM IN INFINITE DIMENSION We shall now reduce the proof of theorem 2 and of Sard-Sm ale's Theorem to their finite-dimensional forms, which are the invariance of Brouer's degree [ 32, 42, 43] for C1-maps and Sard's theorem [ 31, 36]. We observe initially that we may suppose that Y is an open subset of some Banach space 300 ISNARD E, and that ЭХ = 0. In fact, we replace f by i?f: f -1 (U) -> jS(U), where U is an open subset of the domain of some local chart (3 for Y, and JS U £Y - f(3X). Observe that if V £ Y £E is an open ball, then if y 0 and yj are in V we have [ y0 , y J £V с Y, where [yo, yi ] is defined to be the closed interval {ty j + ( 1 - 1)y0 I 0 S t a 1} . So we need only to prove: Theorem 1: Suppose X is a real or complex or, respectively, an oriented C^-Banach-manifold (kê 1), Y is an open subset of some real or complex Banach space E, and f: X -» Y is a proper Ск-Фр-тар, where p is any integer. Then: Sard-Smale-Quinn: If к > p then every у 6 Y is the limit of some sequence of regular values. Main theorem for degree theory: If f is а С^Фо-тар, or а С^-Ф^-тар in the oriented case, and y 0 , уг are regular values for f such that [ y o , yi 1 £ Y, then deg(f, X, y0) = deg(f, X, y x). Proofs: Call К = f" 1(y), or, respectively, К = f"1 ([ y0 > YiD. then К is compact. By theorem 2, which follows, there is some finite-dimensional subspace F of E such that R(f'(x)) + F = E for all x e K, where Definition: For every xE X call R(f'(x)) the range of (fû'"1) '(a(x)), where a is an arbitrary chart in some atlas for X such that xe Ua. Of course, R(f'(x)) is a linear subspace of E that does not depend on a. Theorem 2: Let f: X -» E be а Ск-Фр-тар (p any integer, к S 1), where E is a real or complex Banach space and X is any real, complex or oriented Ck-Banach-manifold. Then, for every F linear subspace of E, the set XF = { x e X I R(f'(x)) + F = E} is open in X. And for every К compact subset of X, there is some finite-dimensional F £ E such that R(f'(x))+F = E for all x in K. Proof: To show XF is open we compose with charts and then we may suppose X is an open subset of some Banach space E0. From the continuity of f1: X -» L(E0, E) it suffices to show that the set of all S e Фр(Ео, E) such that R(S) + F = E is open in L(E q, E), which is proved in lemma 1 of section E. Now, suppose К is compact in X. Then each point of X is in XF for some finite-dimensional subspace F of E because for each Fredholm operator T from a Banach space to E there is some finite-dimensional F £ E such that R(T) + F = E (lemma 1 of section E). The compact set К is then covered by finitely many open sets XF¡, with dim F¡ < +». When we call F the sum of the F¡ we have dim F < +oo and R(f'(x)) + F = E for all x in K. Continuation of proof of theorem 1: We may suppose that the finite- dimensional subspace F of E given by theorem 2 contains y or, respectively, y 0 and у i, adding to F the linear span of those points, if necessary. We may actually suppose that R(f’(x)) +■ F = E for all x e X, because by theorem 2 the set XF of the x £ X such that R(f'(x)) + F = E is open in X, therefore, we may replace Y by Y 0 = Y - f(X-XF) and X by Xo = f" 1(Y0)£ XF (Y0 is an open subset of Y by section C, lemma 1). We have then f transversal to F, according to the following definition: IAEA- SMR- 11/36 301 Definition: Let F Q E be an inclusion of real or, respectively, complex Banach spaces, and X be a real or, respectively, complex Ck-Banach- manifold. We say that a Ck-<6q-map f: X -* E is transversal to F when R(f'(x)) + F = E for all x e f^ F ). When this condition is satisfied we have: Transversality theorem (section E): There is some atlas making f_1 (F) into a real or, respectively, complex Ck-Banach-manifold of dimension q + dim F, in such a way that the restriction f: f_:L(F) -> F is a Ck-$q-map that has as regular points exactly the points of f_:L(F) that are regular for f: X -> E. If, furthermore, f is а Ск-Ф*-тар and X is oriented, then there is one atlas m aking f _1(F) into an oriented Ck-manifold modelled after F, in such a way that the restriction f: f_1 (F) -> F of f is а Ск-Ф*-тар, that has the same orientation at each regular point as f: X -> E. Continuation of proof of theorem 1: It follows that the regular values for the restriction f: f_1(F) -► Y n F of f are exactly the regular values for f: X - Y that are in Y n F. Hence Sard-Smale's theorem for f follows from the corresponding result for f. The same can be said about the main theorem, because [ yo, y il £ Y n F and f and ?" have the same degree at each regular value in Y n F (the same pre-images, with the same orientations in the oriented case). Remark: In finite dimensions the complex case of the Main Theorem follows from the real oriented case: Every complex normed space E corresponds to a real normed space Er equal to E except for the multiplication, which is restricted to real scalars. Each basis vj, . . ., vm for E corresponds to some basis v1( ivx, .... vm, ivm for Er. Every Te L(E, Ej) corresponds to some Tr e L(Er, E lr) with the same norm, and R(T) = E <-> R(Tr) = Er. If T e L(E, E) then det T = | det Tr |2 , as one can see by a computation in Ref. [ 6, p. 47] (done after choice of a corresponding basis such that T had a triangular representation). Since the norms in E and Er are the same, C1-maps from open subsets of E to E become C 1 maps from open subsets of Er to Er, with the derivatives corresponding through the map T -> Tr from LÍE, E) to L(Er, Er). So the regular values are the same for both maps. By induction, we obtain the same statement for Ck-maps. T :E ^ E is an isomorphism if and only if Tr : Er = Er is an orientation-preserving iso morphism because det(Tr) = | det t | 2. Hence, if X is a complex Ck-manifold modelled after E and we replace E by Er in all charts, X becomes an oriented Ck-manifold Xr modelled after Er. And every Ck-map: X -* E becomes an orientation-preserving Ck-map Xt -» Er. E. TRANSVERSALITY Definition of submanifolds: Suppose Y is a real, complex or oriented Ck-Banach-manifold, and suppose a subset M of Y is covered by the domains Ug of some charts /3: Us s (3(Ug ) £ Eg in some atlas for X, such that for each of those 0 there is a closed linear subspace F s of the Banach space Ee such that (3(Ug n M) = /3(UB) n F 0. Then the restrictions ¡3: Ug n M = |3(Ug) n F e form an atlas of class Ck for the topological space M, in the topology from X, modelling M after the Banach spaces Fg. We call then M a submanifold of X, 302 ISNARD and we say that those charts (3 model the inclusion M c Y after the inclusions Fg £. Ee . If, furthermore, that atlas for M happens to be an orientation atlas, which, of course, requires all the Fg to be equal to one fixed F, then M is called an oriented submanifold of X. Remark: When all Fg = 0 we have a О-dimensional submanifold. We shall consider all О-dimensional submanifolds of oriented Ck-manifolds to be oriented submanifolds, by definition. Theorem 1: If f: X -» Y is a Ck-map such that f(Mj) £ M2, where X and Y are complex, real or oriented Ck-Banach-manifolds and Mi and M2 are (oriented or not) submanifolds of X or, respectively, Y, then the restriction Mj -» M2 of f is also a Ck-map. Proof: Taking compositions with local charts, it follows from the particular case where X and Y are open subsets of Banach spaces and Mx and M2 are intersections of X and, respectively, Y with closed linear subspaces. Definition: Suppose f: X -> Y is any Ck-Fredholm-map, with к i 1, where X and Y are both real, complex or oriented Ck-Banach-manifolds, and suppose M is a submanifold (oriented or not) of Y. We say that f is transversal to M on a subset A of X when for every x e A n Г 2(М) there is some chart |3 modelling the inclusion M £ Y after the inclusion Fg £ Eg of Banach spaces, in some atlas for Y, such that f(x) e Ug and (|3 f)'(x) + Fg = Eg [1]. Remark: Any Ck-Fredholm-map is then transversal to any of its regular values (with each value considered as a 0-dimensional submanifold). The next two theorems will be proved in this section (for the degree theory of С 1-Ф *-тар (index 0) it is enough to consider the case T = I, E 0 = E). Pull-back theorem: Let X and Y be oriented Ck-Banach-manifolds modelled after real Banach spaces E 0 f 0 or, respectively, E, suppose f: X -> Y is a Ск-Ф„.(Т)-тар, where T: Eo~* E is any Fredholm operator. Then there is one unique oriented Ck-manifold XT modelled after E, obtained through the attachment of some orientation atlas to the topological space X, such that the identity X = XT is an orientation-preserving Ck-diffeomorphism and f: X -» Y is a Ck-(T + Lc(Eo, E))-map, i. e. the derivatives of the compositions of f with charts in atlases for X j and Y are operators of the type T + compact linear. Definition: XT is the (T)-pull-back of Y by f. An atlas for X T, which we call the pull-back atlas, consists of all orientation-preserving Ck- diffeomorphisms ip between open su b se ts U,, of X and i»(U^) of Eq such that j3fcp'1 is of the type T + map into some finite-dimensional subspace of E, for charts 3 in some atlas for Y such that f(Uv) с Ug. Remark: When T = I, E 0 = E, X j corresponds to the pull-back GLC- structures in Ref. [13], except that now it has a uniquely defined orientation. Remark: Xx = X if X and Y are finite-dimensional because then, if T € L(E0, E), we have T + LC(E0, E) = L(Eo, E) = Ф*(Т); hence any Ck-map X-* Y is а Ск-Ф*(Т)-тар and is a Ck-(T+Lc(E0, E))-map. IAEA- SMR-11/36 303 Transversality theorem (i) Thom's theorem: Suppose X and Y are real or complex Ck-Banach manifolds (k È 1) and f: X -> Y is a Ck-$q-map transversal to some submanifold M of Y (q any in teger). Then f _1 (M) is a submanifold of X of dimension q + dim M such that the restriction?; f" 1(M) - M of f is а Ск-Фч- т а р . (ii) Suppose X and Y are oriented Ck-manifolds (ksl) modelled after real Banach spaces Eo and E, suppose M is an oriented submanifold of Y with the inclusion MÇY modelled after some inclusion F £ E of real Banach spaces, and suppose f; X -» Y is а Ск-Ф.«(Т)-тар transversal to M, where T: E 0 -> E is some Fredholm operator transversal to F. Then f_1 (M) is an oriented submanifold of the T-pull-back XT of Y by f, with the inclusion modelled after the inclusion T_:l(F)£Eo of Banach spaces. The restriction f: f_1(M) -» M of f is a Ck-(T + Lc(T_:l (F), F))-map, where ¥: T‘ 1 (F) -» F is the restriction of T, which is a Fredholm operator of the same index as T. (Definition: Call the oriented Ck-manifold f" 1(M) the T -p u ll-b ack of M by f) . (iii) In both cases (i) and (ii) the regular points for f are exactly the regular points for f that belong to the set f^M ), and the regular values for?"are exactly the regular values for f that belong to M. Furthermore, in case (ii) when T = I and E 0 = E, f and f have the same orientations at each of these regular points. The statement of the theorem is much simplified when X and Y are finite dimensional: Particular case: Suppose f: X -* Y is any Ck-map (ks 1) transversal to some submanifold (or, respectively, oriented submanifold) M of У, where X and Y are finite-dimensional Ck-manifolds (or, respectively, oriented Ck-manifolds). Then f_:l(M) is a submanifold (or, respectively, an oriented submanifold) of X of dimension dim M + dim X - dim Y, and the restriction f: f“1(M) -» M of f is a Ck -map that has as regular points exactly the regular points of f that are in the set f_1(M). Furthermore, if X and Y are oriented and modelled after the same real space, and if dim M / 0, f and f have the orientation at each of those regular points. Theorem 2: An atlas for the inclusion f_1(M )£X of Ck-Banach-manifolds in case (i), consists of all Ck-diffeomórphisms '1 is of the type T Corollary: Taking M ={y} in the transversality theorem, we obtain that if у is a regular value for f then f"I(y) is a submanifold with dimension equal to the index of f, and that in case (ii) this submanifold is oriented in a uniquely defined way. This justifies the definitions, in the introduction, of the degree of proper Fredholm maps with positive index: f‘ 1 (y) = deg(f, X, y). 304 ISNARD Corollary: Sn_1 ={x e Rn | £ x? = 1} is an oriented C“-submanifold of Rn. More generally, let E be any real Banach space such that the function x - |x| is a Ck-map (ki 1) on E-{0} (e. g. any real Hilbert space, with к = +oo). Then the set {x £ E j [ x I = 1} is an oriented Ck-submanifold of E. Proof: 1 is a regular value for the function cp-. x -* |x| ER on E-{0}, because • / \ . x+tx - X I I Transversality property: Suppose f, X, Y, X and Y are as in any of the definitions of the degree (section C), and suppose the restriction X -» Y of f is transversal to a submanifold M of Y or, respectively, to an oriented submanifold M of Y. Then the restriction f: f'^M ) -* M of f is a proper map (because f is so), and if one considers the submanifold or, respectively, the oriented submanifold f_:L(M) П X of X given by the transversality theorem, the restriction f-^M) n x -» M of f”is а С 1-Ф0-тар, or, respectively, С1 - Ф *- т а р . Furthermore, any у GM - f(3X) that is a regular value for either f or f, is a regular value for both those maps. We have also deg(f, X, y) = deg(f^ f'-^M), y), for all ye M - f(9X). The proof of the theorems in this section is based on the following result, which may be regarded as an inverse function theorem modulo F, when one calls a regular point modulo F for a map g: X -► E any x e X such that R (g ’ (x)) + F = E . Theorem 3: Suppose g: X -» E is а Ск-Фо-тар, (kî 1), where X is a real, complex, or oriented Ck-Banach-manifold, and suppose R(g'(x0)) + F = E for som e x0 e X and some F linear subspace of E. Then there is some Ck- diffeomorphism cp between some open subset U^, of X and some open subset of E, such that (cp-g) (Uv) is contained in some finite-dimensional linear subspace of F, and such that x0e U^, and cp(x0) = g(x0). Furthermore, there is one such cp that is orientation-preserving, when X is an oriented Ck- manifold modelled after E, g is а Ск-Ф„.-тар, F f 0 and R(g'(x0)) + F = E. Proof: Taking compositions with charts at x, we may suppose that X is an open subset of some Banach space Eo (which is equal to E the oriented case). By lemma 3 there is some L e L(E q, F) such that g'(xo) + L is an iso morphism Ео= E, which is orientation-preserving in the oriented case. Then x 0 is a regular point for g + L - L(xo), with positive orientation in the oriented case. We apply to g + L - L(x0) the inverse mapping theorem (section C), obtaining U^,, and cp by restriction. For maps of non-zero index we need the following generalization: Theorem 4: Suppose g: X E is а Ск-Фч-тар, where q is any integer, кё 1, X is a real, complex or oriented Ck-Banach-manifold, and suppose IAEA- SMR-11/36 305 R(g'(xo)) + F = E for some x 0 e X and some F linear subspace of E. Suppose T e ФЧ(Е0, E) is such that R(T) + F = E where E 0 is an arbitrary Banach space. Then there is some Ck-diffeomorphism tp between some open subset Uv of X and some open subset of E 0 such that (Ttp-g) (U^ ) is contained in some finite-dimensional linear subspace of F, with xo e ü ,. Furthermore, there is one such tp that is orientation-preserving, when X is an oriented Ck-manifold modelled after E, g is а Ск-Ф*(Т)-тар, T"1 (F) f 0, and R(g'(x)) + F = 0. Proof: By lemma 1, there is some T e Ф_Ч(Е, E0) such that I - TT and I - TT are projectors onto finite-dimensional subspaces of F or, respectively, T_1F, and such that R(Tg)'(x) + T_:LF = E0. By lemma 2 fg is а СкФ*(1)-тар (because when S e Ф*(Т), then TS e ®*(fT) = Ф*(1)). We now apply theorem 3 to íg and obtain Definition: When T e L(E0, E) and F £E we call T + LC(E0, F) = { S £ L(Eo, E) | I S-T is compact and R(S-T) £ F}. Proof of the pull-back theorem: Take M = Y and F = Ej in the following proof of case (ii) (the oriented case) of the transversality theorem. Proof of the transversality theorem and of theorem 2: By theorem 4, the domains U,, of the Lemma 4: In the conditions of theorem 4, we have tp(Uv n g '1 (F)) = = Proof: For xeU,, we have tp(x) e T _:LF -» T( F = F = R(h'(w)) + F _ E R(ÎT'(w)) R ( h '( w )) n F ~ R(h'(w)) ~ R ( h ’ (w)) Hence w is a regular point for h if and only if it is a regular point for h. Suppose now T = I, Eq = E, F / 0 and g is а Ск-Ф+-тар. The E coincide with the determinant of the common restriction F^, -* F,, . Therefore the orientations also coincide. Continuation of proof of transversality theorem: To complete the proof, it suffices to show that in case (ii) the charts Proof of the pull-back theorem: Call CT the collection of all orientation preserving Ck-diffeomorphisms between open subsets of X and of E, such that f Proof that f -1(M) is oriented: Suppose that ue f" 1(M) n n u , / 0 , where each Oif - T Lemma 5: Say 0 ^ F ÍE are real Banach spaces and A = {S £L(E, Е) | I S(F) ç F}. Then A is a Banach subspace of L(E, E) (actually, a subalgebra), and in A are defined continuous linear maps S -* Sr e L(F, F) and IAEA-SMR-11/36 307 S - SqGL(E/F, E/F), where Sr and Sq are the maps induced by S. If S-I is compact, so are Sr-I and Sq-I, and then S is an orientation-preserving, or, respectively, reversing, isomorphism E = E if and only if Sr : F = F and Sq: E/F = E/F are isomorphisms with the same or, respectively, with different orientations. In particular, if S - I has finite-dimensional range, so have Sr - I and Sq - I, and then det(S) = det(Sq) • det(Sr). Remark: If S ^ I + LC(E, E) we may have isomorphism S: E = E such that S(F) £ F: For instance, in E = i 2 ® ^ 2» define S(xi, x2, . . . ), (yi, y2, • • • )) = = ((y1# x x, x 2, . . . ), (y2, y3, . . . )), then S (£ 2 ® 0) t- ® 0. L em m a 6: L et 6: E j E 2 be an isomorphism of real Banach spaces, then for S e L(E2, E2) we have SGGL¡t(E2)~» 0' 1 o S °в G G LJ(EX). Proof of lemma 5:; We have |sr| S j S | and |S q| S |s|, hence S - Sr and S -> Sq are continuous on A. If S is compact, so are Sr and Sq, hence if S - I is compact so are Sr - I and Sq - I. Suppose from now on that S-I, S r - I and Sq - I are compact. Then S £ GL(E) [ Sr e GL(F) and Sq £ GL(E/F) ], because if S GGL(E) then Sr is injective and Sq is surjective, hence, since St e $ 0(F, F) and Sq e ®0(E/F, E/F), we have Sr e GL(E) and Sqe GL (E/F). By chapter III, section B, corollary 4, each path-component of GLC (E) ПА contains one path-component of G Lf(E) n A. Then every S G GLC (E) П A is connected by some continuous path t -» St £ GLC(E) n A to som e elem ent of GLf (E) n A. The paths t -> (St)r g GLC(F) and t -» (St )q g GLC(E/F) are a lso continuous, and the orientations are constant along those paths, so it suffices to consider the orientations when S GGLf(E) n A. But then R(S-I) £Fq finite-dimensional Q E, R(St -I) £ F n Fq, and R(Sa-I) ç^ :+F Fo 4 ; ~ F F 0 n F So to prove det(S) = det(Sr) • det(Sq) it suffices to consider the case where E is finite-dimensional. Then E = F x (E/F), so it derives from: L em m a 7: If E i and E 2 are finite-dimensional vector spaces and S £ L ( E i x E 2,E ix E 2) is defined by S(u 1, u2) = (Siu 1 + Au2 , S2 u2), for S^LfEj, EJ, S 2 e L(E2j E 2) and A e L(E2, Ej), then det(S) = d et^ ) • det(S2). Proof: S = S2§i, where S(ui, u2) = (Siui + Au2, u2), and S2 (щ , u 2) = (u i, S 2(u2)). Then det(S) = det(S2 ) • detiSj ) = det(S2) • det(Sj ), because RfSx -I) £ E 2 x 0 and R(S2 - I ) £ 0 x E 2. Proof of lemma 6 : S -> в'1 S в is an isomorphism L(E2, E 2) = L(Ei, Ei) that has for restriction an homeomorphism GLC(E2) = GLjfEj), which then takes path-component onto path-component. Lemma 1: If T e L(E0, E) and R(T) + F = E, then T G Ф^Ео, E) *-* the restric tion T " 1 (F) -♦ F of T is in $q(T'1 F,F). For each integer q the set {T e $ q(Eo,E) I R(T) + F = E} is open in L(Eo, E). Also, for each T in this set R(T) is closed, E = R(T) © F 0 for some finite-dimensional Fo£F, and there is some Í g Ф.Ч(Е, E 0) such that I - TT and I - ÍT are projectors onto finite-dimensional subspaces of F, or, respectively, of T ' 1 (F), and such 3 0 8 ISNARD that {S e Proof: Suppose T e L(E0, E) and R(T) + F = E. The restriction T'^F) - F is in ФЧ(Т_1 (Р), F) if and only if T £ ФЧ(Е0, E) because both null-spaces are N (T ), and F F _ R (T ) + F dim T(T-iF) dim R(T) n F - dim R(T) Suppose now T e ФЧ(Е0, E) and R(T) + F = E. The finite-dimensional vector space E/R(T) has some basis consisting of the quotient classes of finitely many elements of F, and we call F 0 their linear span. Then F 0 £ F , d im F 0 < +co, and E = R (T ) Д, F 0, where for the moment this direct sum is in the algebraic sense only, 1. e. we have not yet proved that the corresponding projectors are continuous. From dim N(T) < +00 we get [ 20] that Eo = Е ^ Щ Т ) for some Ei which is a closed subspace of Eo, because it is the null-space of a continuous linear projector. Then E j is a Banach space. We observe that the map S: E! xFo -* E defined by S(u, v) = Tu + v is an isomorphism, because by the open mapping theorem [ 20 ] any continuous linear bijection between Banach spaces has a continuous inverse. Then E = R ( T ) ® F 0 is a direct sum in the topological sense, because it is the isomorphic image (by S) of a direct sum. R(T) is closed because it is the null-space of some continuous linear projector. Define T: E E 0 to be null on Fo and to be on R(T) the inverse of the restriction Ex = R(T) of T, then TT and TT are projectors onto Ej or, respectively, R(T) null on N(T) or, respectively, Fo . Hence, I - TT and I - TT are projectors onto N(T) or, respectively, F 0. Suppose, now, that S £L (E 0, E) and E = R(S) + F, then R(T) = R(fS) + ÍF , so E = R(fS) + f(F) + T" 1 (F) = RCfS) + T ' 1 (F). Conversely, ifS Lemma 2: Suppose S €L(E0, Ej) and T eL(E j, E2), where E 0, Ej and E 2 are real or complex Banach spaces. If any two of the operators S, T, and TS are Fredholm operators, then so are all three, and index (T S) = = index (T) + index S. Proof: In Ref. [20]. To complete the proof of lemma 1 it remains to be shown that the set {S e L (E 0, E) I ÍS e Ф0(Е0, E 0), R(TS) + T ’^F) = E0} is open in L(E0, E). From the continuity of the transformation S -> TS it suffices to prove that the s e t {V £ Ф0(Е0, Eo) | R(V) + T_:l(F) = E 0} is open in L(E0, E0). This is p r o v e d in IAEA- SM R-11/36 3 0 9 Lemma 3: If Eo, E are real or complex Banach spaces, and F is a linear subspace of E then the set {T e'$0(E0, E) | R(T) + F = E } is open and equal to the set {H-L | H : Ец s E isomorphism, LeLf(E o,F)} where Lf(Eo , F) = = { L £ L ( E 0,E) j R(L)£F, dim R(L) < +oo}. Also, if E is a real Banach space, T G Ф*(1) £ L(E, E), and R(T) + F = E for some F / 0 linear subspace of E, the.n there is some L £L(E, E) with finite-dimensional range contained in F, such that H = T + L: E = E is an orientation-preserving isomorphism. Proof: The set of the H-L as above is open in L(Eo, E) because it is open for each fixed L since the set of all isomorphisms Eo 3 E is open in L(Eo, E) (because GL(E) is open in L(E, E) [20]). Each H-L as above is in Ф0(Е о. E) because L is compact. Also R(H-L) + F = R(H) + F = E because R(L) £F. Suppose now T e Фц(Ео, E) is given such that E = R(T) + F. Then by lemma 1, E = R(T) ® F 0 for some F 0 £F. From index (T) = 0 we get dim N(T) = dim F 0 , so there is some isomorphism N(T) = F 0 that we extend to some L e L(E0, E), which we define to be null on some Ej chosen so that E 0 = E-l ® N(T). Then T+L is an isomorphism E 0 = E, because its restrictions E x = R(T) and N(T) = F 0 a r e so . S u p p o se T 6 Ф,(1) £ L(E, E) and E = R(T) + F, F / 0. By what was just proved we have then T + L e GL(E) for some L with finite-dimensional range contained in F. Then T + L GGL(E) n Ф*(1), and we may suppose T+L is orientation-preserving, otherwise we replace T+L by (I+C) (T+L) = T + Lj, where С eL(E, E) is such that I+ C eGL¿(E) and R(C) £F is finite-dimensional: Then Lx = L + C(T+L) has range in R(L) + R(C), which is finite-dimensional£F. Such С may be obtained by choosing some one-dimensional F £ F , defining J = I + С to be -I on F and to be I on some E such that E = E © F, then R(C) = R(J-I) = F, and the determinant of J is the determinant of -I: F -> F, w hich i s - 1 . F. HOMOTOPY PROPERTY Throughout this section J and J are subsets of the scalar field (R or C), with J open and J 2. J, and X, X, Y, Ÿ are as in the definitions of the degree. We have, therefore, three cases: the complex case, the oriented case, and the general case of the mod 2 degree. X x J, or, respectively, Y x J is the product manifold, which is complex, or oriented, or real if X or, respectively, Y is so, an atlas for X x J is { o x I | a e atlas for X, I = identity on J}. Y x J is defined similarly. From lemma 1 following it follows that X x J is oriented if X is so. Homotopy property 1 : Suppose (x, t) -* ft(x) is a continuous proper map X x J t extending some Carnap X x J- *Y which is such that for each t e J the restriction ft : X-* Y is а С 1-Ф0-тар (or, respectively, а С 1-Ф*-тар, for the oriented case). Let у be in Y - ft(3X) for all t in some connected subset С of J. Then deg(ft, X, y) is constant for t e C. A map (x, t) -* ft(x) as above is called a proper C1-homotopy of Фо-maps. The theorem has a more general form where both у and the domain of f are permitted to change with t: Homotopy property 2: Suppose Г2 and Г2 are subsets of X x scalar field, with Г2 open in X x scalar field and contained in ñ. Suppose (x, t) -*■ ft(x) is 310 ISNARD a continuous proper map ft -* Ÿ that extends some C1-map Г2 - Y such that for each t scalar, if Gt = { x € X | (x, t) £ is not empty, then ft : Gt -> Y i s а С 1-Фо_тар (or, respectively, а С 1-Ф*-тар, in the oriented case). Suppose t •* yt £ Y is a continuous map on some connected subset С of the scalar field such that yt f ft (x) if (x, t) £ f t - f 2. Then deg(ft, Gt, yt) is constant for t in C. If Gt = 0 for some t in C, then deg(ft, Gt, yt) = 0 for all t in C. Proof: The map q>: ft-* Ÿ x scalar field defined by Lemma 1: Suppose E, Ei and F are real or complex Banach spaces and T £ L(E xF, Ej xF) is of the type T(u, v) = (Îu+Sv, v), where T e L(E, E x) and S £ L ( F , E-l). Then, for any integer p we have: (i) T e i p( E x F , E i x F ) — T е ф р(Е , E i ) . (ii) T is an isomorphism E xF s Ei x F is and only if T is an isomorphism E = E i . When the Banach spaces are real, given some A £ Фр(Е, E x) we have (iii) T 6 Ф*(А xI)«-*T £ Ф*(A) (where (Axl)(u, v) = (Au, v), u e E, v £ F ). (iv) W hen the s p a c e s a r e r e a l, E = E i , then T £ Ф*(1) n G L ( E x F ) *-* T £ Ф*(1)П n GL(E), and then T and T have the same orientation. Proof: (i) and (ii) are proved in section B. To prove (iii), observe that from (i) it follows that for 0 S t S 1 tT + (1-t) (Axl) is a Fredholm operator E x F <-> Ei xF if and only if tT + (1 -t)A is a Fredholm operator E -» E i. To prove the orientations in (iv), we get from section В a continuous path t -* Tt £ Ф*(1) n GL(E) connecting T to some element T x of GLf(E) (0 fi t S 1). We define T t(u, v) = (Ttu + tSv, v), which gives a continuous path t-* Tt £G L (E xF ) П Ф^.(1) connecting T to Tj x I, which has the sam e deter minant, hence the same orientation as T i. Since the orientations are constant along those paths, it follows that T and T have the same orientation. Continuation of proof of homotopy property: For all t 6 С we have then deg( G. THE MULTIPLICATIVE PROPERTY Suppose f: X -» Y and g: Y -♦ W are continuous proper maps that have for restrictions С^Ф о-тар (or, respectively, С 1-Ф%-тар з in the oriented case) X -» Y and Y -* W, where X, Y and W are real or complex C 1-B a n a c h - manifolds (or, respectively oriented C1-manifolds modelled after some real IAEA-SMR-11/36 311 Banach space E), that are open subsets of arbitrary topological spaces X, Ÿ and W. We suppose also that X and Y are Hausdorff spaces. Let { Yj I j} be the collection of connected components of Y - f(9X), and let each y¡ e Yj be an arbitrary point. Then deg(g • f, X, w) = Ç deg(f, X, yj )• d eg (g, Yj, w), for every w e W - g(f(9X) U 9Y). And this sum is finite because we have d eg (g, Yj, w) = 0 for all but finitely many indices j. Remark: gf is С^-Фо-тар from X to W, by section E, lemma 2, but, in the oriented case, it may fail to be а С^-Ф^-тар, because the composition of two operators belonging to I* (I) may be outside Ф*(1). In this case we define the degree of g f on X as the one when X is replaced by Xj = the (I) - pull-back of Y by f (section E), what does not change deg(f, X, yj) for each j and what makes g f а С1-Ф*-тар, because f becomes a CM l + LC(E, E))-map. So, in the oriented case, we shall suppose in the proof that X has already been replaced, so that f is a C M l + L C(E, E))-map. P r o o f: g _1 (w) is a compact subset of Y - f(9X) = y Yj, so g"1 (w) i s co n tain ed in at most finitely many of the Yj. For the remaining Yj we have g ' 1(w) n n Yj = j), hence deg(g, Yj, w) = 0, by the surjectivity property, (g f ) '1 (w) = = f'M g’Mw)) Q yf'M Yj), so degfgf, X, w) = S deg(gf,f_1(Y j), w), by the additivity property. Also, for each j deg(f,JX, yj) = deg(f, f _1(Yj), yj), by the excision property. So it suffices to prove for each j that deg(g f, f‘ 1(Yj), w) = = deg(f, f _1(Yj), yj) • deg(g, Yj, w). In other term s, we may suppose Y = Y is connected and X = f_1 (Y) = X. Then к = deg(f, X, y) is constant for у GY, and all we need to show is that deg(g f, X, w) = к deg(g, Y, w). We may suppose w is a regular value for g, by the invariance property. Then g_1(w) i s a fin ite set. The case g‘ 1(w) is empty is settled by the surjectivity property. Suppose, now, g_1(w) = {уц, . . ., yq}. Each y¡ is a regular value for g, hence it it some open set Vi where g is a C 1-diffeomorphism onto some open set containing w. We may suppose that the V¡ are pairwise disjoint. By the additivity property it suffices to show that for each i deg(g f, f"1^ ), w) = = к deg(g, Vj, w). So, replacing X and Y, we may suppose that g is a C1- diffeomorphism, and it suffices to show that deg(g f, X, g(y)) = = deg(f, X, y) • deg(g, Y, g(y)) for all y e Y. We may suppose y is a regular value for f. Then f_1(y) is a finite set {xj, . . ., xp} = (gf)"1 (gy). So the complex case and the case of the mod 2 degree are proved, and the multipli cation law of orientations settles the oriented case. Remark: When f and g have positive index, the multiplication of degrees is the cobordism ring multiplication of Ref. [ 39]. One needs to show that if a compact connected submanifold M of Y consists of regular values for a p r o p e r С к+1 -Фк-тар f: X -» Y, then f_:L(M) is cobordant to f'4yo} x M, for an arbitrary yo£M , and that in the oriented case (i. e ., M oriented submanifold, X, Y oriented manifolds, f a C k+1 -Ф*(Т)-map), f' 1 (M) and f " 1{yo}x M are oriented cobordant. N o te : P r o o f th at Ф*(1)Г1 GL(Ë) has two path-components (E real). The canonical map L(E.E) q: L(E-E) ■* Е Ж 1) 312 ISNARD has for a restriction a principal-GLc(E)-bundle GL(E) -> q(®o(E, E)) ([ 10, 11, 2 8, 42]). Since the set q($*(I)) is contractible, it is the domain of a local section for that bundle [ 34, 35], which is then trivial over it [23]. This means that the set q"1 (q($*(I))) = Ф*(1) n GL(E) is homeomorphic to q(®*(I)) n GLC(E), which has two path-components (actually, has the homotopic type of G LC(E)). REFERENCES [1] ABRAHAM, R ., ROBBIN, J . , Transversality of Maps and Flows» Benjamin, New York (1967). [2] ATIYAH, M. F ., К-Theory, Benjamin, New York (1967). [3] BROWDER, F .E ., "Local and global properties of nonlinear map mappings in Banach spaces", Istituto Nazionaledi Alta Matematica, Symposia Mathematica, 2 (1968). [4] BROWDER, F .E ., NUSSBAUM, R.D., The topological degree for noncompact nonlinear mappings in Banach spaces, Bull. Am. Math. Soc. 74 (1968) 671. [5] CACCIOPOLI, R., Sulle corrispondenze funzionali inverse diramate, Rend. Acc. Naz. Lincei 24 (1936) 258; 416. [6] CRONIN, J . , Fixed points and topological degree in nonlinear analysis, Math. 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[43] ISNARD, C ., Introduction to the topological degree (to be published, Atas. Soc. Bras. Mat.) [44] ISNARD, C ., A generalization of the Leray-Schauder degree (to be published). IAEA-SMR-11/37 EXISTENCE AND NON-EXISTENCE FOR SEMI-LINEAR ELLIPTIC EQUATIONS J.L. KAZDAN Department of Mathematics, University of Pennsylvania, Philadelphia, Pa., United States of America Abstract EXISTENCE AND NON-EXISTENCE FOR SEMI-UNEAR ELLIPTIC EQUATIONS. After a summary of the linear theory, existence and non-existence of a given semi-linear equation are studied. Several applications are presented. 1. INTRODUCTION Since the existence theory of linear elliptic partial differential equations is in a fairly complete form, one should seriously consider non-linear equations. The question we have in mind is quite primitive: on the assumption that all data are smooth — does there exist, at least, one solution of a given semi-linear equation such as Ди + cu = f(x, u) in fi, with и = 0 on 9fi (1. 1) where Я С Rn is, say, the unit ball and c(x) is some given smooth function? Little is known on this basic question. For simplicity, in this paper we shall limit our discussion to Eq. (1. 1) and give several examples. Most of what follows extends to Lu = f(x, u) where L is any second-order uniformly elliptic operator, not necessarily self-adjoint, and with various boundary conditions, as well as on compact manifolds. A more complete discussion with proofs and additional refer ences is to be found in Ref. [ 6 ] . Э2 3 2 u n Notation: Ли = — r? + . . . + -—я , ÎÎCR° ------Эх^ Эх* is a bounded domain with smooth boundary, and the innerproduct in L 2(fi) is denoted by < , )> . 2. SUMMARY OF LINEAR THEORY The linear equation Ди + cu = F(x) in fi, with и = 0 on 9fi (2. 1) is classical. We list only the facts we shall refer to. 315 316 KAZDAN Existence. If the only solution of the homogeneous equation (F = 0) is u = 0, then given any F there is a unique solution of Eq. (2. 1). On the other hand, if there is a non-trivial solution of the homogeneous equation, then this kernel is finite-dimensional and one can solve Eq. (2.1) if and only if F is orthogonal to the kernel. This is, of course, just a statement of the Fredholm alternative. A simple example is u' 1 + u = F(x) oil Í2 = {0 < x < 7Г} with the boundary conditions u(0) = a(n) = 0. Here sin x is a basis for the kernel of u' ' + u with the given boundary conditions, so a solution exists if and only if 7Г J' F(x) sin x dx = 0 о Although the above generalizes to higher-order equations and system s, what follows holds for the m ost part only for second-order equations. First eigenfunction. The first eigenvalue (i.e. the lowest one) of -(Д|p + с has multiplicity one and the corresponding eigenfunction срг does not change sign, so one can assume 0 on [It is insufficiently well-known that this generalizes to non-self-adjoint elliptic equations by the Krein-Rutman theory of positive operators [ 7, section 6 ] applied to the corresponding compact Green's operator. Then one finds that the eigenvalue with lowest real part is, in fact, a real number and the corresponding eigen function does not change sign. Moreover, Хх depends continuously on Í2 and if a domain u С Q then X^io) > X^fi). Of course, for non-self-adjoint equations the remaining eigenvalues may be complex. ] Extended maximum principle. Assume the constant a < Хг with Xx as above. If u satisfies -[Au + cu] й ou in Q, usO on then either u = 0 or else u > 0 in Q . F o r с s 0 (so Х-l > 0) and a = 0 this is the classical strong maximum principle. As far as we know the extension was first used by Duff [4] for Eq. (2. 1). It has recently been extended to non-self-adjoint second-order elliptic operators by Amann (see Refs [ 1, 8]). 3. PERTURBATION THEOREMS The least surprising results occur when the function c(x) in Eq. (1. 1) is such that one can always solve Eq. (2.1), i.e. when L = Д + c(x)I is 1AEA-SMR-11/3T 317 "invertible". One then anticipates that if f(x, u) is not too large then one can always solve Eq. (1. 1). In this direction the following theorems are v alid : I. If L = A + c(x)I is invertible and | -y | is sufficiently small, then there exists a solution of Lu = y f(x, u, V u) in Г2 , u = 0 on 3!2 (se e Ref. [3, pp. 373-74]). This same proof yields solvability for -y = 1 if instead one assum es that ÍÍ is sufficiently small. The proof immediately extends to higher-order operators and systems. II. If L is invertible, f(x, u,Vu) is uniformly bounded for all xef2, u e C 1 (Я), and g(x, u,Vu) is sufficiently small then one can solve Lu + g(x, u,Vu)u = f(x, u, Vu) in Q, u = 0 on ЗГ2 (3. 1) Here, roughly speaking, one wants to choose g so small that the term gu in Eq. (3. 1) does not get into the eigenvalue region of L, thus making Lu + gu not invertible. To be more specific, if XN < XN+1 are successive eigenvalues of L and if there are constants 7 such that for all x£f2, u e C 1 (f2) < TN s g (x ,u ,v u ) á TN + 1< ^ N + 1 then there is a solution of Eq. (3. 1). This was proved in various degrees of generality for second-order self-adjoint equations by Dolph, Landesman, Lazer, and Leach, and has been extended to non-self-adjoint equations of higher order and to system s by us [ 6] . III. If there is a constant 7 < X-l such that lim su p - S 7 (3 .2 ) |s| - » » then there exists a solution of Eq. (1.1). The motivation here is the special linear case f(x, s) = a(x)s + F(x) for which Eq. (1.1) reduces to the linear e q u atio n Д u + (c - a)u = F(x) which is always solvable if -a s 7 < X j. Existence under the assumption (3.2) was first proved by Stampacchia [9, section 10 ] who also allowed additional non-linearities on the left-hand side of Eq. (1. 1), and was extended to non-self-adjoint equations by us [ 6] . Our proof is sketched in section 5 below. We do not know of any generali zation to higher-order equations or system s. A special case is when с á 0 and f(x, u) = g(x, u) + h(x, u), where one assumes that for all xefi, s£ R I g(x, s) I S const and - 1ц (x, s) S 7 < Xa This can easily be seen to satisfy Eq. (3.2) by the mean-value theorem. The classical case [3, pp. 372-74] is where 7 = 0 . In particular, this gives existence if с á 0 and f u(x, u) ё 0 . One should note that these three conditions for existence of a solution all begin with a linear operator L that is invertible and then place conditions 3 1 8 KAZDAN on the non-linearlity so that the resulting operator does not, in some sense, interact with the spectrum of L. In cases II and III, this allows for certain large non-linearities since the whole semi-infinite interval X < Xj is not in the spectrum of L. If one begins either with a linear operator L that is not invertible or else with large non-linearities f(x, u), then existence is more subtle and essentially unexplored. There are some existence theorems and some non-existence theorems, m ost of which are motivated by the linear theory, i.e. are valid despite the non-linearities. Thus, in practice, given a specific equation, one has to develop new devices. In the remaining sections we shall content ourselves with illustrating various phenomena using rather elementary procedures. 4. METHODS OF PROVING EXISTENCE There are several methods of proving that a solution of a certain elliptic equation does exist: (i) calculus of variations, (ii) sub- and super- solutions, (iii) monotonicity, (iv) continuity method, (v) Schauder fixed-point theorem, (vi) Leray-Schauder degree. All but the second method have been discussed in recent texts and monographs, so we shall only treat method (ii), and this only for Eq. (1. 1). Let us write Lu = Ди + cu. Then the existence theorem for (ii) asserts that if thereis a sub-solution u_, i.e. a solution ofthe inequalities Lu S f(x, и ) in fi , и S 0 on 9fi (4. 1) and if there is a super solution u+ satisfying Lu+ á f(x, u+) in fi , u+ S 0 on 9fi (4.2) and if u. S u+ , then there is a solution и of Eq. (1. 1). Moreover, и satisfies u_ s и á u+ in fi. Thus, in this method, existence is reduced to the — occasionally formidable — task of finding a-priori estim ates in the form of sub- and super-solutions. One decided virtue of this method is that it neither assum es that L is invertible nor does it require any growth hypothesis on f(x, u). Although only recently exploited, this method for proving existence is actually quite old [3, pp. 370-71, where a special — but easily generalized — case is treated] . One can also extend this method to certain second-order quasi-linear equations [ 2 ]. As a simple example, consider the problem of solving Ди = a(x)u - 1 + b(x)ueu in fi, и = 0 on 9fi where b(x) ê const > 0 and a(x) is arbitrary. Existence is immediate IAEA-SMR-11/37 319 upon noticing that we can let u_(x) = 0 and u+(x) = large const > 0. Note that if one rewrote the equations as Ди - a(x)u = -1 + b (x )u eu then it might appear that trouble, i.e. non-existence, would occur if -a(x) was an eigenvalue of Д, whereas our existence proof shows that, in fact, this trouble does not occur. Thus, one has existence because of the non- lin e a r ity . The next two sections will give other applications of this method. 5. SKETCH OF PROOF OF III IN SECTION 3 We first rewrite the hypothesis to read that there is a constant sQ > 0 and a constant 7 < Xx, such that if | s| ê sfl then for all x e f 2, _f(x, _ _ s) s , 7 Next one applies the extended maximum principle to show that since 7 < Хх , then there is a solution v 2 1 in Q o f A v + ( c + 7 ) v S 0 inn, v=const=m on ЭГ2 if m is sufficiently large. We now note that a super-solution of Eq. (1. 1) is u+ = s 0v г s0, since then f(x, u+) 2 - 7 U+, so that Д u+ + cu+ á - 7 u+ g f(x, u+) Similarly, one lets u_ = -u+ to find a sub-solution. This completes the proof that a solution exists. 6 . ADDITIONAL APPLICATIONS Another application is to show that if one can solve Ди = F(x) - g(x, u) in Q, и = 0 on 3f2 (6 . 1) where F is any given function and g(x, s) ш 0 (or g(x, s) s 0) for all seR , then one can solve A v = F (x ) - 7 g(x, v) in !2, v = 0 on ЭГ2 (6.2) for any Os 7 S 1 , To prove this, say g(x, s) § 0 (if not, replace и and v by -u and -v, respectively). We need only find a sub- and a super-solution of Eq. (6 . 2). A super-solution is evidently v+ = u. Let tp be the solution of Aip = F in fi, One can use this method for some problems arising in physics and differential geometry, where one seeks a positive solution of Eq. (1. 1). For this see [ 6, sections 4-6]. A particular and intriguing problem — which is still not fully understood — is when one can find a positive solution to a Ди = au + bu in fi, и = 0 on 9fi w h ere a > 1, a and b are constants, and fi С Rn. In this case, existence depends on the coefficients a, b, on the magnitude of a compared to the dimension n, and on some geometric properties of fi. A different sort of application is to a question posed by Gel'fand [ 5, p. 357 ]. He asked if existence for Eq. (1. 1) for a domain fi implies existence for all smaller domains uCfi. Using sub- and super-solutions, it is easy to see, for example, that if f(x, s) g 0 (or if f(x, s) s 0) fo r all xefi, seR , then the answer is "yes": one can solve E q.(l.l) for any sm aller domain u. For this same question, since if и) С fi then X1 (и) Й A ^ fi) , the condition of III in Section 3 shows additional circum stances under which solvability of Eq. (1. 1) for a certain domain implies solvability for all sm aller domains. On the other hand, the simple linear equation u 1 ' = 1 - и on fi = {0 < x < 4 r } w ith и = 0 on 9fi is an example where one has existence for a domain fi but not for the sm aller domain u) = {0 < x < ir) (see Section 2). Thus, in general, the answer to Gel'fand's question is "no". 7. NON-EXISTENCE The simplest non-existence results use linear existence theory. As a crude start, one might consider Ди + XjU = f(x, u) in fi, и = 0 on 9fi (7. 1) where X} is the first eigenvalue of Д in fi. If a solution и exists, then, by the linear theory, f(x, u) must be orthogonal to the first eigenfunction i-pi (just take the inner product of Eq. (7. 1) with Д и = t F ( x ) eu in fi, и = 0 on 9fi (7. 2) in mind, where т is a parameter and F i s a given function. If т ê 0 and F > 0, then one can always solve this by III of Section 3 (since then f(x, u) = t F ( x ) eu has fu ê 0). However, if т < 0 is large and negative then one has non-existence. This non-existence is shown as follows. First consider Ди = f(x, u) and assume f(x, s) á 0 for all seR , and that f(x, s) s fs(x, 0)s for s a 0. IAEA-SMR-11/37 321 Let Xj be the first eigenvalue of - [ Aq> - fs (x, 0) -Х,<(p, u У = ■yip, Ди - fs(x, 0)u > = (cp, f(x, u) - fs (x, 0)u)> S 0 But this is a contradiction since the left-hand side is positive. To apply this to Eq. (7. 2), one notes that if т is large and negative then the eigenvalue Xj in Eq. (7. 3) is negative. This can readily be extended to a large class of equations. We forego the extension, however, and turn to a more interesting consequence of these results, namely, assuming F > 0 inEq. (7.2), then there is a constant - oo < t 0 < 0 such that one can solve Eq. (7. 2) for all t > t 0 but one cannot solve Eq. (7. 2) for any т < т 0 . The existence for all т § 0 and the non-existence for some тг , sufficiently large negative, were shown above. Existence for т < 0 with |t| small is a consequence of I in Section 3. On the other hand, the first application in section 6 shows that non-solvability for some < 0 implies non-solvability for all т < and completes the p r o o f. Needless to say, this phenomenon also extends to a large class of equations. In addition, by a different but more special proof, one can extend the non-existence for Eq. (7. 2) for т large and negative to certain F of variable sign, namely, if the solution ф o í Аф = F in £2, ф = 0 on 9Q is negative somewhere, then there is no solution of Eq. (7. 2) for т large and negative. We shall close by mentioning some peculiar aspects concerning finding a solution of the deceptively innocent-looking equation Ди = с - k(x)eu (7.4) on a compact Riemannian manifold M (without boundary), see Ref. [10]. Here с is a constant. Note that the linear map Д in Eq. (7. 4) is not invertible since кег(Д) = {constants}. The theory of this equation is fairly complete if с < 0 and dim(M) arbitrary. For с = 0, if dim(M) = 2 then necessary and sufficient conditions for existence are known, while if dim(M) i 3 the suf ficiency is an open question — and is perhaps the m ost simple-looking equation that is yet unresolved. On the other hand, almost nothing is known 322 KAZDAN if c > 0. We mention only what we think is the m ost unusual phenomenon: if M = S 2 (with the standard metric) and if с = 2, then any solution u must s a tis fy ^ e u Vk • VF dA = 0 where F is any first-order spherical harmonic, AF + 2F = 0. In particular, this shows non-existence for Eq. (7. 4) on S 2 if с = 2 and к = F + const. ACKNOWLEDGEMENTS This work was in part supported by NSF Grant GP 28976 X. This paper, in particular, is based on joint work with F.W . Warner. REFERENCES [1 ] AMANN, H .t A uniqueness theorem for nonlinear elliptic boundary value problems, Arch. Rat. Mech. Anal. 44 (1972) 178. [2 ] CHOQUET-BRUHAT, Y . , LERAY, J ., Sur le problème de Dirichlet, quasilinéaire, d’ordre 2, C.R. Ser. A. 274 (1972) 81. [3 ] COURANT, R ., HILBERT, D ., Methods of Mathematical Physics, 2, Interscience-Wiley, New York (1962). [4 ] DUFF, G .F .D ., Eigenvalues and maximal domains for a quasi-linear elliptic equation, Math. Ann. 131 (1956) 28. [5 ] GEL’FAND, I . М ., Some problems in the theory of quasilinear equations, Usp. Mat. Nauk 14 (1959). English translation in A. M.S. Translations (2) 29_(1963) 295. [ 6] KAZDAN, J.L., WARNER, F .W ., Remarks on some nonlinear elliptic equations (to appear). [7 ] KREIN, M .G ., RUTMAN, M. A ., Linear operators which leave a cone in Banach space invariant, Usp. Mat. Nauk £ (1948) 3. English translation in A. M. S. Translation (1) 26 (1950). [ 8] SERRIN, J., A remark on the preceding paper of Amann, Arch. Rat. Mech. Anal. 44(1972) 182. [9 ] STAMPACCH1A, G ., On some regular multiple integral problems in the calculus of variations, Commun. Pure Appl. Math. 16^(1963) 383. [10] KAZDAN, J. L ., WARNER, F. W ., Curvature functions for compact 2-manifolds, Ann. Math. 99 (1974) 14. IAEA-SMR-11/38 AN EXAMPLE OF A STRANGE THREE-DIMENSIONAL SURFACE IN (E2 J.J. KOHN Istituto di Matematica "U. Dini”, Viale Morgagni, Florence, Italy Abstract AN EXAMPLE OF A STRANGE THREE-DIMENSIONAL SURFACE IN C2. After discussion of some well-known geometric properties of pseudo-convex surfaces in (En, an example of a strange three-dimensional surface in Œ2 is described. In this paper, we describe an example which was found by Nirenberg and the author (see Ref. [3]). First we shall discuss some well-known geometric properties of pseudo-convex surfaces in _m L ’ L ai A j=i w h ere (a.lt . . ., am) e IRm and m I aJ rXj (P) = o j=i Here rXj = dr/dxj. Then the Hessian of r at P, denoted by HP(r), is a quadratic form on T P(S) given by H ( r ) ( L ) = £ r XiXj (P) a . a . (1) The following theorem is then known and easy to prove: Theorem A: M is (strictly) convex whenever for each P £ S the quadratic form Hp(r) is positive (definite). Note that, in the special case m = 2 and r(x, y) = y - f(x), the above theorem reduces to the elementary statement that convexity is equivalent 323 3 2 4 KOHN to f" S 0. The general theorem can be reduced to this statement by intersecting S with two-dimensional planes and making the appropriate linear changes of co-ordinates. Recall that convexity of M is also characterized by the existence of separating hyperplanes, i. e. M is convex if and only if for each P é S there exists a linear function which vanishes at P but does not vanish at anv point in M. In the theory of several complex variables there is a notion which is very basic and which is in many ways analogous to convexity. It is called pseudo-convexity and we shall now describe it. Let M с (Cn be a domain with a smooth boundary S given by a real-valued function r as above. Let z i, . . ., zn denote the holomorphic co-ordinates in u y . U z i = * + nT - T u v . u Z j = i (UX: ■M For P e S we denote by T 1,0 the space of complex vectors of type (1, 0) tangent to S at P, if L is any such vector it can be written in the form: L-I*i if: <31 (P) = 0 The complex analogue of the Hessian is the so called Levi form which is a Hermitian form for each P £ S, on the space Tp ' 0 (S), if L e T u (S) given by expression (3) then the Levi form applied to L is given by zjZj (P) a i aj (4) ■ b M is called pseudo-convex if this form is positive semi-definite and strongly pseudo-convex if it is positive definite. It is important (and easy) to check that these notions are independent of the choice of the defining function r (i. e. if r' is another function such that r' = 0 on S, dr' / 0 and r' > 0 outside of M and r' < 0 in M then the number of positive, negative and zero eigenvalues of the Levi form relative to r' at P is the same as that of the Levi form relative to r at P). Furthermore, these notions are independent of the choice of holomorphic co-ordinates. So that just as convexity depends only on the linear structure so pseudo convexity depends only on the complex structure. IAEA- SM R-11/38 325 Observe that the Levi form is an n x n matrix applied to an (n-1)- dimensional subspace. Consider the function R defined by R = e Xr - 1 (5) This function also defines the boundary (i. e. dR / 0, R > 0 outside of M and R < 0 in M). Furthermore, if the Levi form is positive definite and if X is sufficiently large then the n x n matrix (Rz.2. (P)) is positive definite. To see this note that the Levi form being positive definite implies that there exists a constant С > 0 such that ^ г 2.г. (P) ajâj S С |a |2 (6) w h en e v er £ r z. (P) at = 0 (7) w h ere № Uil2 Now we can write (Cm as a direct sum of the subspace of vectors satisfying condition (7) and the one-dimensional subspace generated by the vector (rZl(P), rz 2(P), . . ., ггп(Р)). We wish to show that there exists a constant su c h that ^ RZizj (p ) (a i + (s j + brZj (p)) - const ^ I a t + brz. (P) |2 (8) First note that ^ | a i + brZi(P) I2 = ^ l a j 2 + |b I2 ^ | r z.( P ) |2 (9) The cross term s vanish because of condition (7). Now Rzjzj = (* rz.z. + X2 rz. rg. ) e Xr so that the right-hand side of (8) e q u a ls jj'-^rzizj (P) aiâj + X ^ r ZiZj (P) (a¡b rZj (P) + b r2. (P) âj) (10) + x2 |b|2 X k Zi(P)l2} 326 KOHN Now the cross theorem can be bounded by X (small const |a |2 + large const |b|2) Choosing the sm all constant sufficiently sm all we can absorb the first term in the first term of (10) (by virtue of (6)) and the second term can then be absorbed in the third term of (10) by choosing X sufficiently large. Thus, using (9), we obtain the desired result. The notion of strong pseudo-convexity is, in many ways, analogous to the notion of strict convexity. In particular, the real parts of holomorphic functions are the analogues of linear functions. This is illustrated in the following, well-known elementary theorem: Theorem B. If M is strongly pseudo-convex and if P e S, then there exists a neighbourhood U of P and a holomorphic function h on U such that h(P) = 0 and P is the only zero of Re(h) which lies in M n u. Proof. By the above rem ark we can choose an r so that the matrix (rz.z. (P)) is positive definite. Then we have, by Taylor's theorem, 1 1 r = Re(h) + ^ rz.Zj(P) ZjZj + 0 (|z|3) (11) w h ere h = 2 ^ r 2. (P) zi + X r z¡2j (P) z iZj i ij and |z I = ( I z ! I2 + . . . + |z u|2 )* Since the second term on the right hand side of Eq. (11) is greater than or equal to const |z |2 we see that whenever h(z) = 0 and z is sm all and f 0 then r is positive and hence z is outside of M. Corollary. Under the same hypothesis as above there exists a neighbourhood U and a holomorphic co-ordinate system vx, . . . , vn on U such S n U is strictly convex relative to the linear structure induced by the real and imaginary parts of the vj . Proof: Since (dh) ^ Owe can set vn = h and choose a holomorphic co-ordinate system vj, .. ., vn with origin at P. Then, from expression (11), we see that the Taylor expansion of r, in term s of the Vj, is r = R e(vn) + ^ r v.v. (P) ViVj + 0 ( I v I3 ) (12) Hence r v¡vj(P) = rv¡vj (P) = 0 and since rv.7. (P) is positive definite we see that the real Hessian is positive definite, and the proof is concluded by theorem A. IAEA-SMR-11/3 8 327 The following is also a well-known result in this field. However, the proof is somewhat involved so we do not include it here. Theorem C. If P e S and the Levi-form is identically zero in S n U, where U is a neighbourhood of P, then there exists a holomorphic function h defined in a neighbourhood V of P such that the set of zeros of Re(h) is precisely S nv, Theorems В and С and many other analogies between convexity and pseudo-convexity make it appear plausible that if M is pseudo-convex and p e S then there exists a holomorphic function h defined in a neighbourhood U of P such that h(P) = 0 and such that h does not vanish in U П M. It also seem s plausible that there should exist a holomorphic co-ordinate system on a U such that S n U is convex with respect to the linear structure induced by this co-ordinate system. The example by Nirenberg and the author shows that these "plausible" statements are, in fact, not true. In (D2 consider the surface S given by r = 0 where r is defined by r = Re(w) + I z 112 + 3 [z 12 R e ( z 10) (13) First for P £ S we wish to describe Tp,0(S). L e t Then L tangent to S means arz + brw = 0 Since from (13) we have r w = \ we conclude that L eTp'°(S) if and only if it is of the form: L = a ¿ - 2arz(P) ¿ (15) We also see that rww = r zw = r wz = 0 and r*z = 36 |z |10 + 33 Re(z10). The Levi form on L is then (36 |z |10 + 33 R e (z 10)) |a |2 which is positive except at z = 0. Then we have the following result: Theorem D. Let S be the surface in (C2 defined by r = 0 with r given by (13). Let U be a neighbourhood of (0, 0) and let h be a holomorphic function defined on U such that h(0, 0) = 0. Then there exist two points (zi, Wi) and (z2, w2) in U such that h(z¡, w¡) = 0, i = 1, 2 and r(z1( w i) > 0 and r(z2, w2) < 0. The proof of this theorem is given in Ref. [3]. To conclude we rem ark that the above example was discovered in connection with the study of local regularity properties of the inhomogeneous Cauchy-Riemann equations, see R e f. [2 ]. REFERENCES [1] GUNNING, R. C., ROSSI, H ., Analytic Functions of Several Complex Variables, Prentice Hall, Englewood Cliffs, N.J. (1965). [2] KOHN, J.J., "Local regularity of ^ on (weakly) pseudo-convex manifolds", J. Diff. Geom. 6 (1972) 523. [3] KOHN, J . J . , NIRENBERG, L ., A pseudo-convex domain not admitting a holomorphic support function, Math. Ann. 201 (1973) 265. IAEA-SMR-11/39 A CONTINUOUS CHANGE OF TOPOLOGICAL TYPE OF RIEMANNIAN MANIFOLDS AND ITS CONNECTION WITH THE EVOLUTION OF HARMONIC FORMS AND SPIN STRUCTURES J. KOMOROWSKI Department of Mathematical Methods in Physics, University of Warsaw, Warsaw, Poland Abstract A CONTINUOUS CHANGE OF TOPOLOGICAL TYPE OF RIEMANNIAN MANIFOLDS AND ITS CONNECTION WITH THE EVOLUTION OF HARMONIC FORMS AND SPIN STRUCTURES. The quotient topology of the superspace Л = U x Riem(X)/Diff (the union is taken over all three- dimensional smooth manifolds) does not admit continuous paths between classes represented by Riemannian metrics on non-diffeomorphic manifolds. In this paper, a "very slightly" weaker topology in ^permitting this deficiency to be avoided is introduced. It seems, as was suggested in several papers of J. A. Wheeler, that a change of the topological type of Riemannian manifolds can be used to describe creation and annihilation of pairs of opposite charges, in the non-singular model of electric field (i.e. a harmonic 1-form e on a Riemannian three-dimensional manifold with a handle, such that the dual*e has only a non-zero period around the handle). The related problems are discussed and some examples and open questions are stated. A relation of the new topology in Л with a multifold nature of the spin-structure superspace is considered. This gives a more rigorous outlook on the very suggestive figure o f superspace drawn in Wheeler* s "Einstein's Vision". This paper is a result of the author's efforts to give some mathematical foundation to a rigorous description of the wonderful and inspiring notion of superspace in 'Einstein's Vision' by J.A . Wheeler. INTRODUCTION Let H = (S2, s0) X (S1, t0) be the Cartesian product of S2 and S 1 with distinguished points s 0e S 2 and íqGS1; it will be called a handle. The set С = ({s 0} X S 1) и (S 2 X {t0}) is a sum of one and two-dimensional cycles in the handle H. Let X be a three-dimensional1 C°°-manifold; by M(X), M(X), (X) we denote the open cone of Riemannian m etrics on X, its closure in r °(è s T* (X)) and a superspace M(X)/Diff (X) (see Refs [1, 3]), respectively. In section 1 we give a construction, named a point addition of the handle, of a three-dimensional C°°-manifold X& С — where v denotes a set of para m eters characterizing the construction — such that X & С is diffeomorphic with X #H 1 All results can be generalized to a number of dimensions ф 3; for details, see remark 7. 329 3 3 0 KOMOROWSKI where X #H is the connected sum obtained by removing two open three- dimensional cells from X and H and identification of their boundaries. In section 2, we introduce a homomorphism k , v „ * n ® T ¥ (X))3u ------y e r (®T*(X&C)) such that if geM (X), then gGM (X& C). The identification of every geM (X) with geM (X & C) induces a 'connected' topology in M(X) uM (X & C) and so i n U - < ^ ( X &. C). This identification may be considered as winding of M (X) on Э M (X $ C) (Fig. 1 ). FIG. 1. Winding of M(X) on dM (X& C). Let be the set of all classes of isomorphic (isometric) three- dimensional Riemannian C“-manifolds X. The natural topology of ЛИ i s that of a quotient space, i.e. induced by the canonical projections M(X) ------ But can be also endowed with the topology induced by the projections M(X) U M(X & C ) ------„ОС where M(X) UM (X&C) has our 'connected' topology. In section 3, we define this new topology in and we prove its 'canonical' character, i.e. its independence of arbitrary elements — point additions, etc. —used in the definition. Good points of this topology are exposed in theorem 3 and r e m a r k 4 . We want to emphasize that such results are not obtainable if one is going to deal with all X # H instead of only with all X& C. A connection of our topological concepts with a description of annihilation of charges is considered in section 5. IAEA-SMR-11/39 331 1. THE POINT ADDITION OF THE HANDLE L e t S2 and S 1 be endowed with their natural m etrics. We define subsets U* = {(s,t)eH : d(s, s0) < а } э { s 0} x S 1 U 2 = l(s,t)eH : d(t,t0) < a}d S 2 X {t Q} w h ere 0 < a < тт. F o r e v e r y 0 < a, (3 < it U * U U 2 is a neighbourhood of С in H and 3 U¿, 3 Ug are two-dimensional C~-submanifolds in H. Definition 1. Let £2 be an open neighbourhood of С in H such that 3 S"2 is a two-dimensional C“"-submanifold in H, diffeomorphic with S2. Let к be a diffeomorphism ЭПХ ]0 , 1] Э ( s , t ) ------►K(s,t)eñ\C such that: 1 ) к (. , 1 ) = i d en, 2 ) there exists 0 < a0< it such that for every a, 0 < a < a0, a vector field F on £2\C, defined as F( k (s.t)) (s,t) is transversal to 3 Uj, and outgoing from U¿, i= 1, 2. The pair (Q, к) is called a radial chart on H. The existence of radial charts is easily seen (Fig. 2). Let X be a three-dimensional C°°-manifold and let be given geM (X). If Sfp = Ж (0, p)cTXQ (X) is such that the map exp: 5fp -* X is a normal chart then Жр is called a domain of exp at x 0EX, Let us take a radial chart (S7,k) on H and a domain, !Xp, of exp at x 0e X , where X has a fixed Riemannian metric geM (X). Now, let be given: 1 ) a diffeomorphism у : 3Q-> 3 preserving the positive orientations; 3Í2 and 3.5^ have the canonical positive orientations defined by the Riemannian structures of H and X, respectively, 2) a C”°-function f (Fig. 3): ] 0,1] ->■ [0, + oo[ such that f(]0, i] ) = 0 and f 1 > 0 on H , 1 ], f (1 ) ê p. FIG. 2. Radial chart. 332 KOMOROWSKI FIG.3. Function f. If pf = sup f_1(] 0, p[ ), then f (] 0, pf ] ) С [ 0, p] . Let Q f = к (9fi X ] 0, pf[ ) UC. fif is a neighbourhood of С in H. We define a map v : fif -* X a s v (к (s,t)) = exp f(t) y (s), (s,t)£3iî X ] 0, pf[ v (p) = x, peC Obviously, v is a C"-m ap, It will be called a point addition of С to X at x 0 eX . The next definition explains this terminology. The set {(fi, к), g, SCp, У, f} will be called param eters of the point addition 2 v. R e m a r k 1. v _1 ((exp 5i^)\{x0}) = к (9fi X ] pf[ ) is the biggest subset of fi su c h th at v restricted to this subset — let this restriction be denoted by — is a diffeomorphism onto its image. Definition 2. Let v be a point addition of С to X at xoeX , where X is a Riemannian manifold with metric geM (X). The parameters of v a r e {(fi, к), Жр, y, f}. We define a C^-manifold X & С as follows: 1) X & С is a topological space equal to((X\{x0}) where J? is the sm allest equivalence relation such that (x ~ p) •4------»■ (x = v (p)) for x e X \{x Q}, pefij.; 2) C°°-manifold structure on X& C is given by the canonical maps in : f i f ------► X & C ix : X \ x 0 ------X & C It is necessary to show that so defined X & С is really of class C “. Since (Im ijj) n (Im i x ) = 1п(у _1( ( е х р {x0})) it is sufficient to prove that the map ixo in restricted to v~l ((expJYp)\{x 0}) is a diffeomorphism onto its image. But this restricted map is equal to v0. Thus the rest is contained in r e m a r k 1 . 2 In the following ,if we speak about a point addition of С to a Riemannian manifold X with a metric g CM (X), we shall write { (Q, к), y, f} instead of {(S3, к), g, y , f} . IAEA-SMR-11/39 333 4 G .4 , Construction o f manifold X& C. v The above construction of the manifold X & С lies in the replacement of x0GX by С with its neighbourhood ftf. This may be illustrated by F ig.4, where is dotted and к (dQ X ]|, pf[ ) is hatched. The hatched subsets of X and are identified. It is seen that X & С is diffeomorphic with X#H ; a point addition is a kind of the connected sum. 2. A TRANSPORT OF CO VARIANT TENSOR FIELDS FROM X ONTO X Let у be a point addition of С to X at xQeX . The canonical map ж X & С -»• X defined as V ix (У) - yelm ix ^ „(y ) =■ ^oi^(y). yeto ia is of class C” . Thus we have a monomorphism 7Г*: Г“(® T* (X)) ------► Г“(® T* (X & С)) (1) We shall denote u = тг*и, и £ Г го(® T* (X)). V - 1 и Remark 2. It is seen that ы = 0 on ifj (v (xQ))CX&C. Thus suppш V Clm ix . In other words, r (ix1)* u (ix (У)). y £ Im ix w (У) = . y^Im i* 3 3 4 KOMOROWSKI So, every point addition of С to X at x0e X points out a way of transport of covariant tensor fields from X onto some homologically different manifold. Now we are going to discuss an interdependence of two such transports given by two point additons of С to X at xQeX . L e t v an d v be point additions of С to X at x 0eX . _It follows from Rem ark 2 that the support of ui £ Г ”( 0 T * ( X & C ) ) ( о г £ е Г " ( 0 T * ( X & C ) ) respectively) is contained in Im ix (or Im f^); here Гх: X \{x 0}-^ X^,C is the analogue of ix : X \ {x 0}-> X ^ C. It is obvious that the diffeomorphism фо = ix ° V 1: Im i x ------*■ Im ix is such that , * ~ _ v T $ ¿ u = u on Im ix If the diffeomorphism Ф 0 had an extension to a diffeomorphism from X& С ontoX ^C, then the structures (X^C, u) and (X4.C, &) would be considered isomorphic3. However, there exists — in generaf—but a homeomorphism fr o m Xfy С onto X & С which extends Ф0. Nevertheless, we have the following Fundamental theorem. Let X be a 3-dimensional Riemannian C“ -manifold with a metric geM (X). If v, ¡7 are point additions of С to X at x 0£ X , у £ Г “(® T* (X)) and keN , then there exists a sequence of diffeomorphisms Фп: X & C -----“ X& c su c h th at Ф'1' OJ - (0 ------► 0 n -»■ in the topology of Г к(® T* (X&C)). The proof will be given later. 3. A NEW TOPOLOGY B y .Ж we denote the set of all classes of isomorphic (isometric) three- dimensional Riemannian C“-manifolds. Originally, the term superspace referred to as in Ref. [ 3], now it means ^ (X ) as in Ref. [ 1 ]. In other words, let E be the set of all three-dimensional C“ -manifolds; then U we also define и M (X ) J = x e £ / „ ' D iff 3 V V In applications, u> can be a Riemannian metric on X; then (X & C , w) and (X & C , oj) are ’ metric1 v v v structures _ the metrics are given by the degenerated tensor fields w and и (see remark 2). IAEA-SMR-11/39 335 When we w riteror_/fw e mean that these spaces are endowed with the quotient topologies. For every X eE we have canonical mappings M(X) —Η ^f(X) —Í— and M (X ) — -— ► Z t í(X ) ---- where i's are surjections and j's are injections. Moreover, the map j is a homeomorphism onto its image being a connected component o f, & o r ~j%, respectively. If geM (X), then we denote [g] = joi (g) Remark 3. It is easily seen that joi is an open map. Let us introduce the following denotation s:= (X, g, x, v) where XeE, geM (X), xCX, v is a point addition of С to the Riemannian manifold (X, g) at the point xeX . We define a family (Eg]) of subsets in Л ^([g]):={uu(V a«f) : U£^([g]), VevF([g])} where ^ ([g ]) (or ^ ( I g l ) , respectively), is a base of neighbourhoods of the p oint [ g ] e ..«f (o r [ g] e , respectively). We recall that geM (X& C). We are going to define a new topology in Л taking ([ g]) to state a base of neighbourhoods of the point [ g] e (We shall do that for every point of ) But now we state few properties of so defined family &>% ([ g ] ) . Proposition 1. Let X, Y eE, Proposition 2. Let Sj = (X, g, x, v), s 2 = (X, g, x, iu), w h ere v an d ц a r e point additions of С to the Riemannian manifold (X, g) at xeX , then for every ^ e ^s 2 ([ g] ) th e r e e x is t s W 'e. ([ g] ) such that W 'C . ‘i/ . The proof, based on the fundamental theorem, will be given later. The above two propositions can be combined into Theorem 1. Let X, Y e E and Now we can define a new topology in Let us take 1 ) 9C = {X JagA such that (X^ is diffeomorphic to X g)« = *. (a = I3) (X£E) => ^ X is diffeomorphic to X„^ "(2) 2) IN =b agX}o6A " а set of point additions; ^agxis a point xCXc, g£M(Xa) addition of С to Xa at x£X o, where X a is equipped with metric g. The above two families, âT and IN, determine the following set: S = { s = ( X ,g , X, V ) : X e a r , geM (X ), xeX . i/ velN} â'IN gx ' a’ ° ogx ' a b ' a ' « ’ o-gx allows us to define a topology, , in Л a s fo llo w s: if m e..^ then as a base of neighbourhoods of the point m we take * (m) ={xeX „?/« :^ xe ^ x(m) w h e re 1) X 0 is the only element of 3C such that there exists geM (X a) for which [ g] = m, s gx £ ®áT]N ’ It is seen that in this definition g is not determined uniquely. However, in the light of theorem 1 , our definition of the topology N is not influenced by this ambiguity. The next theorem tells us about a ’canonical’ character of the introduced topology in Л: Theorem 2. Let be given two pairs, with the properties listed in relations (2): ^={ХЛ£А » = 1 v U a x e x a g e m (x„) and ^ = < Y a}oGA M = {^ agy}aeA y e ï a g £ M (ï„ ) T h e n a rN » m Proof. On account of the symmetry of our problem it is sufficient to show that for every m e.ffand each of its -neighbourhoods “Xthere exists anneighbourhood СУ' of m such that Thus we have proved that the new topology ■^s - m in -¿f is independent of the auxiliary elements SC and IN. Its relation to the natural (quotient) topology in is given in the following obvious Theorem 3. The topology is weaker then the natural (quotient) topology in .. R e m a r k 4 . T h e s e t [ M (S3)] U [ M (S2 X S 1)] С i s : 1 ) not connected — has two connected components — if is endowed with the natural (quotient) topology, (2) connected if is endowed with the topology Moreover, let X eLand X(p) = X # (S2 X S1) § Рн.т “ j¡ (S2 X S 1) then U [ М (Х(р))] С „if is connected if is endowed with the topology ■7~аг-ы . p = о 4. THE PROOFS Proof of the fundamental theorem. Let {(Q,k), 5Г , 7 , f}, {(í2,k), 7 , f } be the param eters of the point additions 7 and 7 , respectively. We now have a commutative diagram as shown in Fig. 5. A. Let us notice that if xelm ix film i^, then 1) there exists (s,t)e 3Í2X such that x = ifio k (s , t), 2) x is mapped by Ф0 = i ^ i ^1 onto $o(x) = ixoix (X),J iñ ^'^i'oiñ 1 (x) = iñ o v ^ o v (к (s, t)) = ijj y _1(exp f(t) -y (s)) = % o k (T '1o 7 (s), f'-^f (t)). B. Let r = \ min (pf, pf), U¡ = ix(X\exp )C X&C, i = 1, 2, and V = C U 2. It is seen that UgCUj and V clm ifi (Fig. 6 ). We shall define Фп giving its restrictions to three disjoint subsets U2, V\in(C), in(C) which cover X&C. Thus we state ФП| Т1 = Фо |TI , ®n|. = ifíoiñ 1 I. 1ГЛ ■ Most of the proof is u2 'U2 '>íí(c ) ‘n'0) devoted to a construction of Фп|„. . ,„Ч=’ФП- C. The domain of Ф п is V \ifi (C) = i^o к ( 9 П X ] 0, f _1 (2r)[ ). Since Ф„has to be a bijection (onto X & C) it is necessary that Im -i>n = ¿ ¡ ¡ o k ( 8 Î Ï X ] 0 , f""1^ ) ! ) . T h u s w e m a y d efin e ijj-i оф по 1а « фп instead o f^ . The next few parts of the proof contains a construction of:^n. FIG. 5. Commutative diagram. 3 3 8 KOMOROWSKI FIG. 6. Illustration of item B. D. We put Фп = Фо on U i\U 2. So we have no troubles with a smooth sticking of Фп and Фп|и2. Let us notice that U i\ Ü2 = in o к (9 Í2 X [ f _1(r), f " 1( 2 r ) [ ) . H e n c e , f o r (s, t) e 9Г2 X [ f _1(r), f _1(2r)[ we have ^П° ‘ !s.t) = «(ï'10T(s)l f _1of(t)) (3) E. The map к (or к,respectively), generates on f¿\С (orfi\C , respectively) a vector field F (or F, respectively), the flow of which is xT(/c(s,t)) = к (s , t + t ) (o r x T (k(s, t)) = k'îs, t + t ), respectively). Thus, we have к(s, t) = X^jÍKfs, 1)) = Xj.jis) and the analogical expression for к'. Now (3) has the fo r m Фп ° K (s,t) = Х~_г ( y -1 о 7 (s)) f о f(t) - 1 F. We shall make use of the following lemma: Let Ebe the set of all such functions fe C “ (IR1) that 1 ) f = 0 on ] - oo, 0 ] 2 ) f 1 > 0 on ] 0, +oo[ (Hence, for every jeN , there exists a¡ > 0 such that f® is non-decreasing on [ 0, o-j ] ). Lemma 1. Iff, feE, 0 It follows from this lemma that there exists a sequence Fne C“(] 0,1]) su c h that 1 ) for every n eN F n =f on [ f‘ 1(r), 1 ], ^ 2) for every neN thereexists t> ^ such that Fn = f on ]0,t], 3) II Fn - f II Ck + 1(]0,ll) nZ, +„* 0 where f, f are param eters of the pointadditions v and v, respectively. We define a sequence une C°°( ] 0,1 ] ) IAEA-SMR-ll/39 339 Х ( Э П , Е 0) aft an А, с FIG. 7. Illustration o f item G . As the next step in the construction of ip w e put ^noK(s,t) = X (7"1o 7 (s )) = k (y _1o 7 (s), un(t)) for (s,t)e9Q X f" 1(2r)[ , what is compatible with relation (3). G. Since к, к are radial charts on H there exist two open neighbourhoods, Ai and Аг, of ССГ2П£2 such that 1) A2 CA2, ЭАх^ЭА2 = Ф, 2) aAjCK (3Í2 X ]0,|[ íníc^ñ X ]0,i[), _ 3) for every p eA x and a, /3 ê 0 ( a +j3 0) = ► (ttF(p) + |3F(p)^ 0), where F, F are vector fields generated by к and /Г, respectively (see definition 1). Let a, sTeC“ (H, [ 0,1 ])be such that r 0 , р е С A j a (p) = 1 L 1 . p e A ^ r 0, p e A a(P) = I 1, p e C A and a+a> 0 onH, ___ L e t в be the flow of the vector field aF + aF defined on ÍL (Fig. 7). It has the following property: XT(P) if P, xT(p)eCA XT(P) if P, xT(p)eA 2 Let eQ > 0 be so sm all that к (ЭГ2, e0)cA 2. For every s'eBft the curve t -* 0T(s), defined for non-positive t , intersects the set к (8Q, e0) in exactly 3 4 0 KOMOROWSKI one point (because the flows в and x are equal on A2). Thus we can define a C“ -function b: 3Q -» IR1 as follows: if se3fi, then b(s) is such that (s) (T_1° T (s ))(S i'i, e0). We have from G.2) that b < - |. Hence, there exist C°°-functions vn such that r “ n ( t ) - l , | s t < f " 1(r) ЭГ2 X ] 0 , f _1(r)[ Э ( s ,t ) -* vn ( s ,t ) = j so m e th in g , e0< t < \ b ( s) - t , t = e0 - t , т е [ 0, e 0[ and 3 vn/3t > 0 . As the next step in the construction of фп we put фпок ( s ,t ) = e Vn(s,t) ( 7 _10 7 (s)) (6) for (s,t)e0íí X ] e0/2 , f _1(2r)[ , what is compatible with definition (4). R e m a r k 5. фпOK (3a I) = = х4(эп) = к(эп, è) H. We introduce the following maps: if (s,t)e3fí X ] 0, f -1 (2r)[ then % (K(s,t)):= eVn(lit) (^ o T lsJle io S X ]0,Г-!(2г)[). Obviously, in the light of definition (6), we have фп = ipn on к ( 3 X ] e0/2, f'^ r)! ). Comparing the definition of R e m a r k 6 . For every 0 ё т < e0the set к (3Q, е 0 -т) is mapped by The last equality follows from (5). I. For every t e ] 0 ,1 ] the map 3Í2 3 S ->• к t(s) = к (s, t) е к ( 3Q, t) is a diffeomorphism. Let re[ 0, eQ[; we define a map d Q 3 s ------orT (s) = K¡l.TocpnoK4. T ( s ) e 3 f t It follows from remark 6 that: 1) Im стт = ЭГ2; 2) crT does not depend on n; 3) it does not depend on t, i.e. crT = tr0 fo r r e [0 , e0[ . The explicit formula fo r aQ i s 3 Q 3 s ------% (s) = K¡0lo e b(s) (7 '1o7(s))e3n It follows from the properties of у and у and from the construction of the flow в th at ct0 is an orientation-preserving diffeomorphism of 3£2. Since ЭГ2 is diffeomorphic with S2, ct0 is smoothly isotopic with idan (see Ref. [ 2] ). In other words, there exists an C“-isotopy IR1 X 3f2 3 (t, s) -*■ Gt(s)£3f2 such that , IAEA-SMR-11/39 341 Finally we put i¡/noic(s,t) = Kt oGjOk”1 о <рпок. (s, t), for (s, t) e9fi X ] 0, f "\ 2 r ) [ , what is compatible with (6). S in c e r cpn on к (ЭП X ] f _1(2r)[ ) Фп = ] е 0 L i d fi on к (9fi X ] 0 , — [ ) the m ap Фп = i~otpnoi~^ gives, with defined in B. diffeomorphisms Ф„| T, and Ф J . , a diffeomorphism from X& C onto X&C. U2 nlliî(c) r v v J. Thus we have constructed the diffeomorphisms ^¡X&C-'-X&C. Now we are going to show that the sequence Фп satisfies the hypothesis of our theorem. It is enough to prove that if ш еГ^Т * (X)), then <ï£ u - & n_^ ^ ► 0 in the topology of Г^Т * (X & C)). 1. Wehave fromB. and D. that Фп = Ф0 on U^. Hence, on Ux, Ф,„ to - ¿J = Фо0 ( ^ х ) * ш - (i'x1) ^ = (i x ) * u - (i'x1) * « = °- 2. From Remark 2 we know that u = 0 on in (к (ЭП X ] 0, j ] ) u C ) = 0 u = 0 on i~ (к (ЭП X ] 0, |] ) UC) = 0 But rem ark 5 tells us that Фп(9<9) = 90, hence Фп(<9) = &. T h u s = 0 - 0 = 0 on 0 . 3. Let us denote W = inoK__(8f2 X ] \, f _1(2r)[ ). Since С (UiU0)cW , it is sufficient to show that (Ф*и -i5) |w p j * 0 in the topology of r k(T* (W)). But i fi0 K m aps ЭП X ] 1, f_1(2r)[ diffeomorphically onto W. So, we shall prove that (in OK)* (^Ц ~Ц ) n^«, “ 0 in the topology of Гк(Т* (9fi X ] j , Г\2г)[ )). Let X be the exponential map 9УГХ ] 0, 1[ Э ( x , t ) ------► X (x,t) = exp ( t x ) e X where .Sfis the unit ball at 0 in TXQ (X). From the definition of ш and ш and from the fact that Фп = i nonW we get (ifiOK)* (Ф*ш-и) = ((Х~1о7офпэ к)* - (А.'1 о у o k ) * ) (X* и) w h ere Х~1о7офпок =:a n an d X_1oi/ok =:a m a p 9fi X ] }, f~\2r)[ diffeomorphically onto th e ir im a g e in Э-5ГХ ] 0, 1 [; h e re Х * и е Г ”(Т * (3УГХ ] 0, 1[ )). Besides en(s.t) = Íy-^TÍS). f°un(t)) = (7_1o7(B). Fn (t)) a(s.t) = Cy ’ ^ Y (s ), f (t)) It follows from F that an — j - a in the topology of the space C k + 1(9fi X ] j , f *(2r)[, 9XX ] 0, 1[ ). Hence, we have obtained that (a * -a*) (X*u) 0 in the topology of Гк(Т* (9fi X ] i , f -1(2r)[ )). 342 KOMOROWSKI FIG. 8. Proof o f proposition 1. Proof of proposition 1 (Fig. 8) We begin with few properties of this diagram. Letк (s.tjefy; then ço/uok ( s ,t ) = (fiiexp^. f(t) ( Г Ф (3Y(y)) = ix o 1 Ф (3Q (P)) = i n (P) , peiif This definition is correct because if jY(y) = j^ÍP). then ix ° 0 * g = & (?) (see Eq. (1)). Indeed, тг‘ (<¡p* g) = (j'1)* qr g = (.pof1)* g = (i*^ )* g = Ф" ((iy’f g ) = Ф" I- Now we pass to our proof. Let us take U и (V ) e ([ g] ) = с^>2 ([ h] ), w h e re U e -Ml g]), Ve3T([ft]). We have to find U 'e .Ж ([ g] ) and V 1 e Л ([ g] ) — th en U 1 U (V 1 n . ) e ^ Sl(f Si ) ~ such that U'u(V'n^)CUu(Vfl„#) (8 ) W e sta te U 1 := U (9) Thus we have only to determine V 1 . At thebeginning of section 3 we have introduced the canonical maps IAEA-SMR-11/39 3 4 3 It follows from the continuity of joi that W = (joi)-1 (V) is a neighbourhood of ft in M (Y &C). S in c e ф is a diffeomorphism, ф* is an isomorphism. Thus,_making use of relation (7), we get that (ф'1)* (W) is a neighbourhood of g€M (X & C). Hence^from rem ark 3, V = [ W] = [ ft//"1)* (W)] is a neighbourhood of [ g]e„ Proof of proposition 2. Let us take U U(Vn„ U'u(V'n.. W e sta te U':= U ( U ) There is only V 1 to be determined. The maps M(X&C) — Η ►„‘i r(X& C) --Í—► J i 7 M M were introduced at the beginning of section 3. _It follows from their proper ties that (joi )"1 (V) is a neighbourhood of ge, «*g-g||k0< Ÿ (12 ) If w e denote W'-= 0 (g, k 0, и °м— ), th en Ф* (W) С 0 (g , k 0, -£■). T h u s, we N 0 obtain from (12) that ÿ:‘ (W )C0(|, ко, ^). Besides, since W is a neighbourhood of g e M (X fe C), it follows from rem ark 3 that [ W] is a neighbourhood of [g]e^ Remark 7. The previous results can automatically be generalized for the case of n-dimensional Riemannian С -manifolds, where n is such that every orientation-preserving diffeomorphism of S n_1 is isotopic with identity. If this is the case we shall obtain this generalization by thereplacement of S 2 by S " ’ 1. 5. THE NEW TOPOLOGY IN „ The electric field (or flow of electric field, respectively) generated by a pair of opposite charges can be described by a harmonic 1 - f o r m (o r 2- f o r m , 3 4 4 KOMOROWSKI respectively) on a three-dimensional Riemannian C“-manifold with a handle. This concept is briefly described in Ref. [3], chapter 2, section 11, and, extensively, in Ref. [4]. According to this idea we have the following correspondences: pair of charges (+q, -q) -«------•- handle in a three-dimensional in a universe Riemannian manifold X #(S 2 X S 1) with a metric g .4 flow of electric field -•------► ееГ (AT* (X #(S 2 X S 1))) su c h that generated by this pair 1 . Ag e = 0 (harmonicity), 2. if C 2 is a 2-cycle on X # (S2 X S 1) around the added handle then4a 3. if is a 2 -cycle non homologous to C 2 then c2 attraction of charges -«------► process in which the distance between charges ('ends' of the handle; see footnote 4) decreases. These correspondences suggest that if one wants to describe the annihilation of pair of charges (+q, -q) one has to introduce such a topology in Л for which such an evolution of m etrics (continuous curve in Л) would be possible that leads to vanishing of the handle. Moreover, on account of the attraction process, this vanishing must consist in the disappearance of the handle, but not in its breaking (Fig. 9), as was suggested in Ref. [ 5] and by several of W heeler's figures in Ref. [3]. Such a new topology in „4? was introduced and its canonical character was proved in theorem_2. The basic idea consists in the identification of each g eM (X ) w ith geM(x&C)?M(X#(S2x s1)). Thus, an arena where the annihilation of charges may take place, is partially prepared. Now, if we want to describe the annihilation, we have to be able to pass continuously from harmonic 2-form s on X § (S 2 X S 1) S' X & С to harmonic 2-forms on X. One can try to do that by an analogous identification of every e e Г(ЛТ*(Х)) with ё e Г (AT* (X & С)) ~ Г ( T* (X # (S2 X S1))). Let us take a c u r v e ] - 1 , 1 [ B t ------► (gt , et ) 4 A physicist may demand the Riemannian manifold (X $(S2 x S1), g) to have the handle. S2 x S1, with a 'very small radius1 added, i.e . inf HC2 « radius o f electron, where II II g is the length in the sense of the metric g and inf is taken over all 2-cycles in X $ (S2 x S1) that go around the added handle. Then a concept o f localization of charges can be introduced because not the whole o f the handle is accessible to our physical (metrical) experiments — we are larger than the electron! 4a Since now we write Г instead of Г 00, however, we shall only deal with C°°-objects. IAEA-SMR-11/39 3 4 5 FIG. 9. Disappearance and breaking of a handle w h ere (gj, et)£M (X & С) X Г (AT* (X&C)) fo r t < 0 (g t, e )eM (X) X Г(ЛТ* (X)) for t ё 0 and (gt. e ) e0)eM(X&C)X Г(Ат*(Х&С)) By the above identification our curve is continuous. Let C 2 be a 2-cycle on X & C going around the handle added to X. We assume that A Rt et = 0 for a ll t an d that J et I = 1 for t < 0 (unit charges). Then, from continuity, c2 we have that / lim e t = lim/ et = 1 J t-» о t-> о J c2 c 2 Hence, we get a contradiction because, according to rem ark 2, we can choose such C 2 that ё„ = 0 on C2. This proves that the proposed identification of e and e — the trick used in the construction of the new topology in -4? — does not lead to such topology in U Г(ЛТ* (X)) ХЕЕ D iff that would be good for a description of the annihilation of charges. (Here E is the set of all three-dimensional C“ -manifolds. ) How does the flow of electric field change when two opposite charges are coming closer and closer? Physical experiments tell us that the flow a) increases between charges; b) decreases to zero at points far, in comparison with the distance of the charges, from the charges; c) the total flow rem ains constant. 3 4 6 KOMOROWSKI FIG. 10. Open set U containing the handle. Thus, we may suppose that if we have a harmonic 2-form e on a three- dimensional manifold X (with a handle) with a Riemannian structure g and e has the only non-zero periods on 2 -cycles going around the handle, then we can choose an open set U cX containing the handle5 (Fig. 10) and a con tinuous curve (0, 1] Эе - (ge, ee)€M (X) X Г (ÀT* (X)) such that (g1( ej) = (g, e), for every е е (0 , 1 ] e £ = 0 and the periods of ee are the sam e as those of e, e W lim i , , 0 e_:= lim e = 0 6 -> о 6 x \u An example of such curve e -* (g£, ee) will be given later. Thus, returning to the problem of a topology in 2 U Г(ЛТ* (X)) X £ E ______D iff we may formulate the following rem ark: 2 , Rem ark. It seem s that one has to identify every ееГ(Л Т* (X)) with a set ë + E„ C Г ( А т * (X&C)) Г(ЛТ (X # (S2 X S1))), where E„ consists of all sections with support on the added handle. Precisely speaking E„: = | е е Г ( Л Т $ (X&C)) ^uppeCfl-ÿ 1 (x0)}/c f. p .5/. г ______Let us give a few definitions: F 2 (X):= M(X) X Г(Ат * (X )), F 2(X): = M(X)X Г(ЛТ*(Х)), F z:= u F 2(X), F ^;= U F 2 (X). The isometry relation x e c X £ L in U M (X) ( U M(X), respectively) induces an equivalence relation in F 2 хег XÊE (or F2, respectively), as follows: if (g¡, e;)eF 2(X i) (o r F 2 (X¡), respectively), i = 1 , 2 , then (gj, ej) ~ (g2, e2) if and only if there exists a diffeomorphism 5 We say that U contains the handle if the quotient map X -» X/U kills the homology elements generated by the handle. IAEA-SMR-11/39 3 4 7 It follows from the previous considerations that we are going to look for descriptions of annihilations among lifts, to ip'2, of curves in „¿f; h e r e , Л is endowed with the new topology. Thus the quotient topology in S ' 2 is too strong for our programme. We can define the weaker one as it was suggested in the above remark. The identification of every (g, e ) e F 2 (X) with {(g, e + e) : eeE „ } C F 2 (X & C) induces a 'connected' topology in F 2 (X) U F 2 (X & C). As the weaker topology in .S'-2, we take that induced by the canonical projections: F 2 (X)UF2 (X&C) V w h ere F 2 ( X ) u F 2 (X& C) is endowed with the above defined 'connected' topology. (The correctness of this definition of the topology in ^ — its independence of a choice of v 's — can be proved as in theorem 1 .) Even when J ^2 is endowed with this weaker topology we still do not know whether it is a proper arena for the description of annihilation processes. We have to give, at least, one example of annihilation as a lift, to S'-2, of a curve in ЛС. (We need not take account of 'equations of motion' of such processes since only kinematics is considered.) This leads us to a problem whose solution is unknown to the author. On account of Hodge's, Kodaira's and de Rham's theorems in the theory of harmonic form s, it will be easier to deal only with compact manifolds, i.e. we take E as the set of all compact three-dimensional C"-m anifolds. If X is a compact 3-dimensional C“ -manifold with a handle then we have the m a p 6 M (X)Bg -> e (g)eT(ÁT* (X)), where e (g) is the unique 2-form such that A ge (g) = 0 and for every 2 - c y c le C 2 1 if C 2 is going around the handle {0 o th e r w ise __ Let be given an open set UCX containing the handle; we define Мц (X)-'= IgeM (X) : g = o). Let Рц Ье the set of all continuous curves (0, 1 ] Э e -2-*- ge e M (X) such that lim geeMu (X). Now we can formulate e-> о Open Problem 7 I. How can be characterized the elements gePjj for which (*) supp lim e (g )CU ? II. Let X - Y & С and U = X \Im iy (see previously), heM (Y). Does there exist an element gePy such that V lim g£ = heM yjX) (see previously) and (*) is fulfilled? 6 It was shown in Ref. [ 6] that this map (generalized for arbitrary dimensions o f X) is o f C 1-class in the sense o f Ref. [7 ] and, o f course, in the sense o f [ 8]. 7 This problem, after straightforward reformulation, can be stated for an n-dimensional compact C°°-manifold X, g€ G M (X ), e (g e) — a harmonic к-form, on X, with prescribed periods, U — an open set in X such that the quotient map X -* X/U shrinks all к-cycles on which the periods o f e (g£) do not vanish. 3 4 8 KOMOROWSKI The following two examples may be useful for better understanding of our problem. Let g be the canonical Riemannian metric on S 2C1R3 an d dt ® dt be the canonical Riemannian metric on S 1 = И 1/Х. Let X = S2X S 1 and U, Uj, U2be as shown in Fig. 11. In the next we shall use the same symbols to denote forms on S 2 o r S 1 and the form s on X induced by them. E x a m p le 1 . Let g£ = r£ g + (r£)2 dt ® dteM (X), where ee(0, 1 j and r£eC°°(X) is such that f £ fo r X e u r (x) H TT € s r 5 1 on U, e ' ' [_ 1 fo r X e U 2, e 1 and lim г бС^Х), Thus the curve e -* ge b e lo n g s to Р ц . e o 3 Let us take e = — * g l g2 , where is the Hodge operator on S 2 induced by the metric g. It is easy to calculate that * e = -Д- dt. Thus e is a harmonic & J ge 47Г 2-form on X and / e = 1. So e(g ) = e for every e e (0, 1 ] . But supp s2 lim e (g ) = supp e = X, hence (v) is not fulfilled. e->0 — ------E x a m p le 2. Let AæS1, i = 1, 2, be open sets such that U A ¡ X SJCU. Let s2 We define g£ = r£ ip2g + (r dt ® dt, where reC"(X) is such that . f e2 fo r x e U „ 1 TT r. (x) = i i , _TT e ¿ á r S — on Ц and lim г, с £ С (X). е ' ' {_ J fo r x e u 2, e 1 о Thus the curve e -* g e belongs to P 0 . L e t usta k e e £ = (**) we get that e (g c) = e£. It is easily seen that supp lim ef C A jX S 1 CU. € “** 0 Hence the condition (*) is fulfilled. IAEA-SMR-11/39 349 6. CONTINUOUS CHANGES OF SPIN-STRUCTURES In the present section E will be the set of all three-dimensional oriented C“ -manifolds and by diffeomorphism we shall mean an orientation-preserving diffeomorphism; we shall denoted them by DiffQ. L e t X e E , geM (X ) then the set S(g) of all spin-structures on the oriented Riemannian manifold (X, g) is defined as the set of those elements of H 1(F(g), Z2) which, restricted to any fibre F(g)x, xeX , give a generator of the cyclic group H^Ffg),;, Zg)8; see Ref. [ 9] . Here F(g) is the principal bundle of oriented g-orthonormal fram es tangent to X; hence H 1 (F (g)x, Z 2 ) = H 1 (SO(3), Z2) = Z2. (It is obvious that for every XeE, geM (X) the set S(g) is non-empty; see footnote 8 .) If g, heM (X ) then the Schmidt orthonormalization is an isomorphism F(g) -» F(h) which induces a bijection S(g) ->• S(h). If we take g0eM (X) then we have a bijection M (X) X S(gQ) -» U S(g). Using this bijection we g £ M (X ) can introduce a topology in the set и S(g). This topology does not depend g e m (X) on the choice of gQ. In the next, we shall denote by S(X) (or F x,respectively) the set S(g0) (or the bundle F(g0) respectively), where gQ is any chosen element of M(X). By the set of all spin-structures on an oriented manifold X eE we shall mean the topological space M(X) X S (X), where S(X) is a discrete space. L e t X , Y e E a n d U M (X) X S (X) ______D iff 0 which would have properties analogous to those of the 'new 1 topology in Let us begin with two lemmas: Lemma 2. If U is a contractible subset of X eE such that X \U is an open submanifold, then the canonical map Fr : S (X) -» S (X \ U), where i :X\U -* X , is bijective. Proof. It suffices to compare a quantity of spin-structures on X and on X\U; see Ref. [ 9]. Lemma 3. If U, V are open submanifolds of X eE such that UDV is connected and H 1(UnV, Z2) = 0 then the canonical homomorphism X = F f X F f , X : h!(Fx, Z2) — * H^Fjj. Z2) © H!(FV, Z2) w h ere i v ; U -*■ X, iy : V -*■ X, gives a bijection X : S (X) -»■ S (U) X S (V) 8 It is known [10 ] that if X is a three-dimensional oriented manifold then T(X ) is a trivial bundle. This fact w ill not be used except in remark 4' ,2). The author wants to proceed a way which would be useful in generalizations for dimensions9^3. 3 5 0 KOMOROWSKI Proof. Let us take the following diagram J tfju n v , Z.) I 0 - h !(F x, z 2) - Hi(FU( Z„) Ф H 1 (F , Z.) 4 H1(F П F , ZJ = H1(F , Z J Г' Н ^БО СЗ), Z 2) where the horizontal sequence is a part of the Mayer - Vietoris exact sequence and the vertical one is an exact sequence extracted from the spectral sequence of the fibration Fun v - Obviously, X (S(X))CS(U) X S(V). On the other hand, if (cr, a')eS(U ) X S(V) then i,;‘o/u (cr, a') = 0 and since i* is injective we get that ц. (cr, cr') = 0. Hence there exists peH 1 (Fx , Z2) such that X(p) = (a, cr'). But a, cr' are spin-structures, thus p must belong to S(X). Let у be a point addition of С to X eE at a point x0eX (see section 1). It can be seen that an orientation of the manifold X & С is determined by the orientation of X. Thus X ^ C e E . v We are going to define a surjection a,, : S (X & C) ------► S(X) V L e t ((fi, к), g, S>f , y, f} be the param eters of v. We define Vr = îrÿ^XXexp Х г)Сл & C, where 0< r 5 pf, then 7r„|Vr is a diffeomorphism onto its im age. Let U = Im iQ С X & С, then Vr U U = X & С and Vr П U is diffeomorphic to S2X I. Let us take the following commutative diagram S (X & C) — S (U) X S (Vr ) where X is the bijection defined in Lem m a 3, second factor, j : V X & С is the natural injection and ■ X \ exp STr — =— ► X By Lemma 2 the map F.,,! is a bijection. The above diagram defines a io TTy surjection a v : S (X & C) -* S (X). It can be seen that a u does not depend on r appearing in the definition. (Let us note that U was not used in the above definition of au. ) Proposition 3. If creS (X) then a 1 (cr) is a 2-element set. Proof. It follows from the above diagram that the number of elements in <*"1(cr)is equal to the number of elements inS(U). Using the Lem m a in Ref. [9] we obtain that it is equal to the number of elements inH 1(U ,Z2) ^ H^S 2 x S 1, Z 2) ^ Z 2. IAEA-SMR-11/39 351 The surjection au was defined for a chosen point addition v . Its independence of this choice may be expressed as Theorem 4. Let X e r b e endowed with a Riemannian metric geM (X). If v, are point additions of С to X at x 0€X , и€Г"(Х T* (X)) and keN then there exists a sequence of (orientation-preserving) diffeomorphisms Ф : X & С ------X & С n и ¡r su ch that Ф * u - u ------П —> oo »■ 0 in the topology of Г к(® T* (X)) and for every n the diagram S(X) is commutative. Proof. As the diffeomorphisms Фп we shall take those constructed in the proof of the fundamental theorem. It is easy to see that they preserve the orientation. To prove the commutativity of our diagram, let us define a (or a~, respectively) using Vt = w'J (X \ exp 5Tr ) (or \^r = 7r~^ ( X \ exp respectively), where r = \ m in (pf, pj"). In section 4B, the set Vr was denoted by Uj. Let us notice that ФП(УГ ) = Vr and the diagram X is commutative. Thus we get commutativity of S(X) which completes the proof. Now we can pass to the construction of a topology in. Let (g, ff)CM(X) X S (X); by [g, a] we denote the class in . containing (g, ст). We shall use also the following notation: if © CM(X) X S (X) (or & С M(X) X S(X)), respectively) then [ & ] is the set of all classes in ,0C U M(X) X S (X) s . X € E ------——------, respectively)containing the elem ents of О. s ¿ЛПл 3 5 2 KOMOROWSKI Let s - (X, g, x ,v ) be as in section 3. We define the following family of subsets in л? : S ^([g,d)=|[DX{cr}]u([VX {ст',а"}[ n „<*;) : Ue.-f(g), Ve Щ where -^(g) (or JV ^^respectively) is a base of neighbourhoods of the point geM (X) (or geM (X), respectively) and {ct'.ct"} = a'J- (ct); se e proposition 3. For so defined ([ g,cr]) we can get an analog of theorem 1. In its proof we have to make use of theorem 4 in the place of the fundamental th e o r e m . Proceeding as in section 2, we take ^"and IN, where E consists of oriented manifolds, and via we define On the basis of the abovementioned analogue of theorem 1 , we can obtain theorem 2 in its spin- structure version, i.e. for the above defined topology in, Theorem 3'. The topology is weaker than the natural (quotient) topology in However, if X eE then both topologies restricted to [M(X)X S(X)]C„ R e m a r k 4 '. 1) If X eE th en X(p) = X#(S 2 X S 1)#. . . #(S2X S^G E and U [M(X(p))X p times p = 0 S(X(p))]c,^ is connected when , ^ is endowed with the topology -^ IN* 2) Let^Z (S 3(p)) = ^ ( S.3(P ^ - Sl S-i(Eil and .^ 0(S 3 (p)) = ] • Then the canonical projection (S3 (p)) -* a (P+ l)-fold c o v e rin g . Proof. 1 ) follows directly from the definition of :7¡jrjNa d 2 ). L e t X eEthen F x = X X SO(3) (see footnote 8 ). L e t ct0 be the generator of H 1 (SO(3), Z 2). We shall denote also by a 0 the induced element of HX(FX, Z2). Thus a 0eS(X ). It is seen that S(X) ={ста = ст0 + т*» : aeH ^X , Z2)}, where тг : Fx -* X. Let us note that if ^eD iff 0(X) then (F*ctq, = ag ) « = ► (q?fa = /3). Now we take X = S 3 (p). Lettt= (« 1, -...Op). (3 = (Эр ...,|Зр) е ф Z 2 = Hi(S3(p), Z2). It can be proved that ( H This follows from the fact that if Ux, . . ., Up are (small) disjoint balls on S 3 then for every аеп(р) there exists an orientation-preserving diffeomorphism cp such that H 1(S 3(p)) = ® z 2 D iff0 П (p) where on the right-hand side we have a set of p + 1 e le m e n ts . Remark 4' is illustrated by Fig. 12. 9 nip) is the group of permutations of p elements. IAEA-SMR-11/39 1 1 FIG. 12. Diagram illustrating remark 4'. ACKNOWLEDGEMENTS I should like to express my gratitude to Professor K. Maurin for his interest in this work and to Dr. W. Szczyrba for an elegant proof of le m m a 1 . REFERENCES [1 ] FISCHER, A.E. , The Theory of Superspace, Relativity, Plenum Press, (1970) p. 303. [2] SMALE, S., Diffeomorphisms of the 2-sphere, Proc. Am. Math. Soc. 10 (1959) 621. [3] WHEELER, J.A ., Einstein's Vision, Springer-Verlag, Berlin (1968). [4] WHEELER, J.A ., Geometrodynamics, Academic Press, N.Y. (1962). [5 ] WHEELER, J.A . , Particles and Geometry, Relativity, Plenum Press (1970) 31. [6 ] SZCZYRBA, W. , Harmonic forms and deformations of metrics on Riemannian manifolds (unpublished). [7] SZCZYRBA, W. , Differentiation in locally convex spaces, Studia Math. 39 (1971) 291. [8 ] OMORI, H. , On the group of diffeomorphisms on a compact manifold, Proc. 1968 AMS Summer Inst, on Global Analysis_15 (1968) 167. [9 ] MILNOR, J. , Spin structures on manifolds, L'Enseignement mathématique (1962) 198. [10] STEENROD, N ., The Topology o f Fibre Bundles, Princeton Univ. Press (1951). IAEA-SMR-11/40 A NEW PROOF FOR REGULARITY OF SOLUTIONS OF ELLIPTIC DIFFERENTIAL OPERATORS M. KURANISffl Department of Mathematics, Columbia University, New York, N. Y ., United States of America Abstract A NEW PROOF FOR REGULARITY OF SOLUTIONS OF ELLIPTIC DIFFERENTIAL OPERATORS. A proof, which avoids use of the mollifier and, instead, employs the power of the Laplacians and the notion of index, is given for the regularity of the solutions of elliptic differential operators. 1. Let L be a partial differential operator. Assume that Lu = v, v is of class C~, and that u is of class Ck. If L is elliptic and coefficients are C°°, it follows that u is of class C°°. Usually, we prove this by means of Sobolev norms and the mollifier, or by constructing a param atrix by means of pseudo-differential-operator theory. Here, we give a proof which uses, instead of the mollifier, the power of Laplacians and the notion of index. 2. Let T be a bounded operator of a Hilbert space H 1 to a Hilbert space H2 . We say that T is an operator with index if the kernel of T and the cokernel of T are finite-dimensional. If this is the case, the index T is defined to be dim ker T — dim coker T. We need the following two propositions: Proposition 1. Assume that T is an operator with index. Let К be a compact operator of H 1 to H2. Then T + К is an operator with index; and the index of T is equal to the index of T + K. Proposition 2. Assume that H1 = H2 and T is Hermitian. Then, the index of T is zero, provided T is an operator with index. 3. We list here properties of Sobolev's norms which we need. Let F be a vector space. In practice, F will be the vector space of functions of class С over a compact manifold, or, more generally, the vector space of sections of class С of a vector bundle over a compact manifold. For e a c h s e R , the field of real numbers, we assume that a pre-Hilbert space inner product (, )s on F is given. We set |]u||s = ((u, u)s)i We denote by (F, s) the pre-Hilbert space F defined by the norm || ||s. Denote by Hs, or HS(F), the completion of (F, s). Hs is a Hilbert space, where the inner product (and the norm, respectively) is also denoted by (, )s (resp. || ||s). 355 356 KURANISHI Assumption (A): The identity map i : (F, s) -*■ (F, t) is continuous for s > t. If this is the case, i induces a unique continuous map i: Hs -*Ht for s > t. Assumption (B): i: Hs -* Ht is injective for s > t. Thus, we may regard Hs as a vector subspace of Ht andconsider the vector space H = USHS. Assumption (C): F = nsHs. L e t F 1 and F 2 be vector spaces on which are given pre-Hilbert spaces satisfying the assumptions (A) to (C). Let L : F 1 - F 2 be a linear map. We say that L is of order m, when L: (F1, s) -* (F2, s - m) is continuous. If this is the case, L induces a bounded linear map L: Hs (F 1) -» Hs_m(F 2) and hence a linear map L: HfF1) - H(F2) 2 3 If L': F -* F is of order m', it is clear that L' o L is of order m + m '. The regularity property in this general context is formulated as follows: We say that L has the regularity property, when L u e F and u E H fF 1) im plies that u e F 1. To formulate our conditions under which the regularity property holds, we have to introduce further assumptions. Assumption (D): There is a pairing map H © H 2 USHS © H.s -» Œ = the field of complex numbers by which H-s becomes the dual Hilbert space of Hs. To be more precise, denote by <^Uj + u 2, w)> = ^u-pW^) +<^u2, w)> , ( « u , w У = a ^u , w У for all u1( u2, ueH s, weH.jjO'eŒ. M o r e o v e r I where cs is a constant, and for any v e H s there is a unique w eH _s su ch th at < u ,w > = (u, w )Q for all u, w e H0. IAEA-SMR-11/40 357 L e t F 1 and F 2 be vector spaces on which are given pre-Hilbert spaces satisfying the assumptions (A) to (D). Let L : F 1 - F 2 be a linear map of order m. A linear map of order m L * . f 2 _ F i is called the adjoint of L if H s( F 1) 3 u-* ( L u , w)> = <(u, v)> fo r a ll uE H sfF1). We set L*w = v and check L* is the adjoint of L. Denote by L s the bounded map H ^F 1) -* Hs.m(F2) induced by L. Then the adjoint of L s (as a bounded map of H JF 1) -*■ Hs.m(F2)) exists, say, (Ls)*. (Ls)* is a map of Hs_m(F2) -» H ^F1). We note here that (Ls)* is different from L*. The form er is based on (, )s and (, )s_m, but the latter is based on <, X If L 1: F 2 -*• F 3 is of order m 1, we see easily that L * 0 L 1* = ( L 1 о L ) * Assumption (E): There is a linear map т ‘ : F -» F of order t satisfying the following conditions: i) r 1: H s+t (F)- H ,(F ) is bijective and bicontinuous, ii) r1: Ht (F) - H 0(F) is an isomorphism of Hilbert spaces, iii) t 1 о тг - rt+r (and (t1)* - r4, respectively) is of order t + r - 1 . (resp. of order t - 1 ). Assumption (F): The injection HS(F) -* Ht(F) (s > t) is a compact linear m a p . 3 5 8 KURANISHI By a Sobelev structure of F we mean a collection {(, ) К.,У, t 51} satisfying assumptions (A) to (F). Let M be a compact manifold and E be a vector bundle over M. By means of a Riemannian metric on M, we define a pseudo-differential o p e r a to r t $: C°° (M,E) -* C“ (M,E) such that its principal symbol is multi plication by | | | s for a cotangent vector Ç. The Riemannian metric on M together with a Hermitian metric on E defines an inner product ( , ) on C“(M, E). Define (, )0 by К..У . Then we define (, )s in such a way that assumption (E) ii) holds. Then we see that these form a Sobolev structure on C” (M,E). L e t F 1, F 2 be vector spaces with Sobolev structures. A linear map L : F 1 -» F 2 is said to be a homomorphism (of order m) of Sobolev structures when it is a map of order m and t sL - L t s is of order m + s - 1 for all s in IR. If L: F 1 -» F 2 is a homomorphism of order m, we see easily by assumption (E) that L*: F 2 -» F 1 is also a homomorphism of order m. I f L 1: F 2 ^ F 3 is a homomorphism of order m ', L 1 o L : F 1 F 3 i s a homomorphism of order m + m'. In the case of Sobolev structures on c ” (M,E) any partial differential operator (or, more generally, pseudo-differential operator) is a homomor phism of Sobolev structures. 4. Our main purpose is to prove the following regularity theorem: Let L be a homomorphism (of order m) of Sobolev structures of F 1 to F 2 . Assume that (*) Il L u U0 + IIuHh,.! ё с II u ||m ( u e F 1) (**) l|L ,;su ||0 + И v ||m_1 * с II v||m ( v e F 2) for a constant с > 0. Then L has the regularity property. We divide the proof of this theorem in several steps: 1 2 i 1) Let L: F -*• F be a homomorphism of order m. Assume that (w) holds. Then for any s С IR (-)s II L u ||s + Cs {J U ||s+m-l - C [J U ||s+m (u £ F ) for constants Cs > 0. Applying this to L*, we see that the assumption (**) im p lie s ( * * ), H L * v ||s + С,- H v ||i+m. 1 g c H v ||s+m (v e F 2 ) Proof. Il Lu II s = ||tsL u ||0 ê II L t s u ||o - ||[ts, L ] u ||0 ê CIIT4 L - lk su||m_1 - Il [TS,L] U II 0 Sin ce t s (and [ ts, L ], respectively) is of order s (and order s + m - 1, respectively), it follows that ||xsu Hm-i + || tTs, L] a ||0 s C s || u Hs+nj-! . 2) Denote by ker L n Hs the kernel of L: HsiF1) -* Hs.m(F2). ker L n Hs is a closed Hilbert subspace of H ^F1). ker L П Hs clearly increases with s. If (*) holds for L, then ker L П Hs is finite dimensional. IAEA-SMR-11/40 359 Proof, ker L n H j being a closed subspace of H ^F1), we regard it as a Hilbert space. By assumption (F), the injection ker L n Hs ->■ ker L П H ^ is a compact linear map. On the other hand, by (which is valid fo r a ll v e H jf F 1) by continuity), we see that the norm || ||s and || on ker L n Hs are equivalent. Then, by a theorem in Banach spaces theory, it follows that the dimension of ker L n Hs is finite. 3) Denote by Ys the orthogonal complement of ker L П Hs. Assume that (*) holds. Then for a constant c^ > 0 llII Lu||s ms й с' s iillull Ms + m '(u £Y s + m ).' In particular, the image of L: H ^ ^ F 1) ->■ HS(F2) is a closed subspace of HS(F2). We denote the image by LH s+m. Proof. If the estimate does not hold, we can find a sequence U yEH ^^F1) such that uveY s+m II Uv||s+m = 1 , ¡I L u v||s 0 Since i: H ^ ^ F 1) -* H j^ .^ F 1) is compact, we may assume that uv is a Cauchy sequence in H ^ ^ ^ F 2) by replacing it by a subsequence if necessary. Since (*)s holds for u e H ^ I F 1) by continuity, it follows that uv is a Cauchy sequence in Hs+m(F1). Thus uv converges to an element u in H ^ ^ F 1). L u = 0 This is a contradiction (because Ys+m n ker L = {0}). Therefore c ’s a s above exists. Once the estimate is established, it is easy to see that LHs+m is a closed subspace of HS(F2). 4) Set X s = { v e H s(F2) ; ( v , z ) = 0 for all z £ ker L* n H_s} Assume that (**) holds. Then Proof. Denote by Y' the orthogonal complement of ker L * n Hs in HS(F2). By 3) applied to L*, we see that L*H_s+m = L*Y ls+m is a closed subspace o f H . J F 1). PickveXs.m. Since for any u e L*H s+m there is aunique y e Y.s+m with u = L*y, we can define the linear map 4>- L>:'H-s+m Э u = L *y « = < y , L ( t ’ s )* t s u'> = w h ere u = ( t ' s )* t"s u'e H.s+2s (F1) = H jiF1). Thus 5) A s s u m e th at F 1 = F 2 = F, L satisfies (*), and that L* = L + R, where R is of order m - 1. Then dim (Hs_m/LH s) = dim (ker L n Hs ) Proof. Consider the bounded linear map L : Hs - H s.m which we denote by L s. By our assumption L satisfies (*) and (**). Hence by 2) and 4), L s is with index. We have to show that the index of L s = 0. Since r ‘m: Hs_m -> Hj is a homeomorphism of topological vector space, the index of L s is equal to that of t -™ Ls: Hs - H s We denote by (r‘mL )* the Hilbert space adjoint of the above map. In view of propositions in 2 ., our contention is proved if we show that (r'mL s)* = r"m L s + К when К is a compact operator. For u, vG Hs (u, (T‘mL s)*v)s = (r'mLu, v)j = <т“ r"mLu, rsv> = = (u, (r5)’ 1 ((T')'1 )*L*(T-,n)*(T,)V v )1 H en ce (T-mL , ) * v = ( t T 1 ( ( t s)"1 ) " I / <(t‘ !:; ) ' (ts rV -v On the other hand, assumption (E) and our assumption in 5) imply that the right-hand side of the above is equal to T‘mL + К where К is of order -1, q. e.d. 6) Under the assumptions in 5), ker L n Hs and ker L* n Hs are independent of s. In particular, ker L n H , and ker L* n Hs are in F. Proof. By 4) dim Hs_m/LHs = dim (ker L* n H.s+m). Hence dim (Hs.m/LHs) increases with s. On the other hand, dim (ker LnHs) decreases as s increases. Then we find by 5) that dim (ker L n H s) is independent of s. Since ker L n Hs Q ker L n Ht for s > t, it follows then ker L n Hs is inde pendent of s. Then ker L n Hs c F by assumption (C). Since L* also satisfies the condition in 5), we see ker L * л Hs is independent of s and ker L* л Hs c F. 7) Under the assumption in 5) the regularity property for L holds. IAEA-SMR-11/40 361 Proof. By 4), LHS = X s_m. On the other hand, ker L* nH s = ker L* n Г by 6). T h u s L H S = ( т е H s_m; < v , z > = 0 fo r a ll z e ker L* n F} By looking at the right-hand side of the above, we conclude that L H t n H s.m = L H S for all ts s. Then, in view of 6) we easily see the following: L u £ H s.m implies that u e H s. The regularity property is a consequence of this conclusion. 8) Let L be a homomorphism (of order m) of Sobolev structures of F 1 to F 2. Assume that L satisfies the conditions (*) and (**). Then the regularity property holds for L. Proof. Lu = v implies L*Lu = L*v. Hence the regularity property of L follows from that of L *L . On the other hand, we easily see that the assumption in 5) for L *L (thus, m there should be replaced by 2m) follows from the condition (*) and (**) for L. The elliptic partial differential operators of order m are known to satisfy conditions (*) and (**). However, Sobolev norms satisfy conditions (A) to (F) only when we consider functions over compact manifolds. There fore, by the above conclusion, we see that elliptic differential operators on compact manifolds have the regularity property. However, we can obtain the local regularity theorem out of the global regularity theorem. To make the matter elementary, let us say that, for a domain П in Rn, a measurable function u belongs to Hs(f2) if and only if fu e H s(Rn) for any C°°-function f with support in a bounded domain where is diffeomorphic to a ball. Let L be an elliptic partial differential operator of order m on Í2. Assume that u e H s(Q), s a y s Ш m, and Lu = v is of class C°°. Now, we assert that u is of class C". Namely, choose f as above, and embed fix in a compact manifold M of dimension n, say, a torus. Construct an elliptic partial differential operator, say L 1; on M which coincides with L for points in K, where К is the support of f. Since Lfu = fv + [ L, f] u and [L,f] is of order m - 1, it follows that Lj^(fu) eH s.m+1 (M). Hence, by the global version, fu eH S+1(M). Therefore, u e H s+1(£2). Then, by induction, we can conclude that u is of class C“ . SECRETARIAT OF SEMINAR ORGANIZING COMMITTEE M. D o lc h e r Istituto di Matematica, Universitá di Trieste, Italy J . E e lls Institute of Advanced Studies, Princeton, N. J. , United States of America J. C. Zeeman Institute of Mathematics, University of Warwick, Coventry, W arks, United Kingdom EDITORIAL BOARD OF PROCEEDINGS P. de la Harpe Institute of Mathematics, University of Warwick, Coventry, W arks, United Kingdom M. D o lc h e r Istituto di Matematica, Université di Trieste, Italy J . E e lls Institute of Advanced Studies, Princeton, N. J. , United States of America J. C. Zeeman Institute of Mathematics, University of Warwick, Coventry, W arks, United Kingdom E D IT O R S A. M. Hamende International Centre for Theoretical Physics, Trieste, Italy J.W. W eil Division of Publications, IAEA, Vienna, Austria 363 HOW TO ORDER IAEA PUBLICATIONS Exclusive sales agents for IAEA publications, to whom all orders and inquiries should be addressed, have been appointed in the following countries: U N IT E D KINGDOM Her Majesty's Stationery Office, P.O. Box 569, London SE 1 9NH U N ITE D S TA TE S OF AM ER ICA UN IPUB, Inc., P.O. Box 433, Murray Hill Station, New York, N.Y. 10016 In the following countries IAEA publications may be purchased from the sales agents or booksellers listed or through your major local booksellers. Payment can be made in local ■ currency or with UNESCO coupons. 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