Durham E-Theses
Characterisations of euler n-spheres
Clark, D. E. R.
How to cite: Clark, D. E. R. (1970) Characterisations of euler n-spheres, Durham theses, Durham University. Available at Durham E-Theses Online: http://etheses.dur.ac.uk/8843/
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Academic Support O ce, Durham University, University O ce, Old Elvet, Durham DH1 3HP e-mail: [email protected] Tel: +44 0191 334 6107 http://etheses.dur.ac.uk Characterisations of
Euler n-Spheres
by
D.E.R. Clark
A thesis in partial fulfilment of the degree of Ph.D. in the University of Durham.
June 1970
•OIEN0F 8/ I - 6 JUL 19719700 * ACKNOWLEDGEMENT S
I am very pleased to express here my sincere gratitude to my supervisor Professor T.J.Willmore. His continued help, guidance and encouragement have been invaluable throughout my studentship.
The author is also grateful to him for reading through the manuscript.
I would also like to thank Dr. B.B.Smyth, Dr. J.P.Wilson and
Mr. K. El Hadi for many profitable discussions. I am grateful also to Dr. B.B.Smyth for bringing to my attention the fact that theorems A and B have simultaneously and independently been proved by Professor Y. Katsurada [32] in Japan.
The receipt of a Science Research Council studentship during the course of this work is gratefully acknowledged.
Finally I wish to thank Mrs. J. Gibson for typing the manuscript. CONTENTS
Page Summary Introduction Chapter I § 1. Smooth Manifolds 1 §2. Multivectors and Forms 3 § 3- The Cartan Calculus h § h. Change of Frame and Global Forms 8 § 5. Submanifolds 10 §6. Some Definitions 15 Chapter II § 1. Standard Equations 25 § 2. Generalised Classical Curvature Theory 27 § 3« Canonical curvature theory 32 Chapter III § 1. The Generalised Minkowski Formula kO § 2. The Second Generalised Integral Formula § 3» Specialisations of I 53 Chapter IV § 1. Characterisation in Codimension one 62 § 2. Characterisation in Arbitrary Codimension 77 § 3» Generalisations 83 Chapter V § 1. Algebraic Preliminaries 88 § 2. Geometric Preliminaries 90 § 3. Integral Formula 93 § k. Specialisations 98 Conclusion 101 Bibliography 10k SUMMARY
The purpose of the present work is to give a multidimensional generalisation of the Liebmann-Suss theorem by means of integral formulas. In order to achieve this it was first necessary to extend the classical local curvature theory. This was done by using the mean normal curvature vector (as a canonical cross-section of the normal bundle) to define generalised second and third fundamental forms from which curvature invariants could be obtained. Secondly, using these invariants, we derive a multidimensional generalisation of the classical integral formula of Minkowski.
As a final application of the integral formula technique we obtain an integral formula for a volume-preserving diffeomorphism between two compact immersed submanifolds. Using this we find conditions under which a diffeomorphism is an isometry. This generalises similar work of Chern and Hsiung. i
INTRODUCTION
One of the most interesting classical results in global
differential geometry is the
Liebmann H-theorem [1]: The only ovaloids of constant mean curvature
H in Euclidean space E3 are the spheres.
In 1901 Hilbert [2] gave an ingenious proof of this result and further proved the
Hilbert K-theorem: A closed surface of constant Gaussian curvature
K in E3 is a sphere.
These theorems characterise the sphere in E3. In 1929 the
H-theorem was generalised by W. Suss [3] for the case of convex n-dimensional hypersurfaces (assuming no self-intersections) in En+^.
In global differential geometry there are available three
1 classical' methods of proving general uniqueness theorems: 1. the
'Index method1, 2. the Maximum method, 3» the Integral formula method. The most effective tool used so far for closed strictly
convex two-dimensional surfaces has been the Index method; but this
does not readily generalise to higher dimensions.
The Maximum method is primarily the work of E. Hopf lh];
however in a long series of papers in 1958 A.D.Alexandrov [5] extended
the method to show that convexity is unnecessary for the validity of ii
the Liebmann-Suss theorem. Proofs up to 1958 have primarily used the elementary symmetric functions of the principal curvatures: however Alexandrov weakened the convexity condition by using more general functions.
The principal difficulty in using the third method is the derivation of the integral formulas themselves, - the actual proof which follows from them is mostly routine. Recently Hsiung [6] and
Feeman and Hsiung [8] have used this method to further extend the
Liebmann-Suss theorem for hypersurfaces embedded in a riemannian manifold of constant curvature. Hsiung was also able to weaken the convexity condition to a 'star-shape1 condition.
The Liebmann-Suss theorem is closely related to the classical integral formulas of Minkowski. It is thus one of our main aims to
develop multidimensional analogues of the Minkowski formulas for arbitrary co-dimension, and hence investigate whether the above mentioned theorems are special cases of more general theorems.
The essential difficulty in the translation of the known methods
of global surface theory to multidimensional differential geometry is
the formal generalisation of the second and third fundamental forms
for codimension greater than one; for each normal vector to an
embedded submanifold gives rise to a corresponding second and third
fundamental form as well as to corresponding curvatures - in particular the r^*1- mean curvature, corresponding to these normals. iii
The work of Chern 19 ], Hsiung [7 ], Schneider [12], Gulbinat [15] and Stong [l!+] clearly indicates the necessity of extending the known local theory of submanifolds and, in particular, to define curvature invariants which depend only on the Normal bundle of the embedding and not on the selection of special cross-sections of the normal
"bundle. The canonical second and third fundamental quadratic forms defined, from which we obtain such curvature invariants, are meaning• ful analogues of the second and third fundamental forms of classical surface theory. The third fundamental form can be interpreted simply as the metric of the 'spherical image'.
By analogy to the classical relation between the second and third fundamental forms it is possible to derive simple relations between the canonical fundamental forms and. the global metric of the submanifold. The aim of this local part of the work is to help clarify the relationship of these forms to the intrinsic geometry of the submanifold.
It turns out that with suitable specialisations and restrictions many of the recent generalisations of the Minkowski formulas can be derived as special cases of our generalised integral formulas.
Further, by restricting ourselves to co-dimension one we obtain the results of Suss [3], Hsiung [ 7]and Yano 0.5 ] directly. .
In I963 Chern and Hsiung [17], using integral formulas, obtained iv
conditions that a diffeomorphism between two compact submanifolds in Euclidean space should be an isometry. We extend their result to the case where the ambient space is an arbitrary riemannian manifold.
In Chapter I we develop the necessary analytic basis for our work, and for the first part of this chapter we depend mainly on the paper of H. Flanders [18], In Chapter II we develop a local
curvature theory of the mean normal curvature vector which overcomes the disadvantage of the earlier theory due to K. Voss [19],
K. Leichtweiss [20] and W. G-ulbinat [13] which depended on the
choice of special cross-sections of the normal bundle. In Chapter III we derive the generalised Minkowski integral formulas; and in
Chapter IV we apply them to give a global characterisation of riemannian
n-spheres and their higher co-dimensional analogue Euler n-spheres.
In Chapter V we generalise the isometry theorem of Chern and
Hsiung [17].
We conclude by indicating possible lines of future research
in this area. 1.
CHAPTER I
In this chapter we develop the basic differential-geometric theory of isometrically immersed submanifolds of a connected riemannian manifold.
§ 1. Smooth Manifolds
Let Mn be an n-dimensional differentiable manifold of class
C°°, i.e. on M we assume an infinitely differentiable structure. Let
F(M) denote the space of all C°° real-valued functions on M. A tangent vector to M at a point p e M is a real-valued function v : F(M) -» ffi satisfying: a) v(f + g) = v(f) + v(g) f,g e F(M) b) v(af) = av(f) a e ]R c) v(fg) = v(f)g(p) + f(p)v(g).
The set Mp of all tangent vectors at p e M forms an n-dimensional
1 n vector space called the tangent space at p. If Ur(x , .x ) is a local coordinate neighbourhood of p, then the vectors e^, l^i^n, at p defined by: e^f) := (df/chc1)^ form a basis for M^. Any tangent vector v at p can thus be represented uniquely by v = vle^ where v1 are unique constants.
The dual space M* of M is called the space of one-forms at P P 2.
p, and it is easily seen that the basis of M* dual to the basis
1 {e. J} of M is dx , .... dx" . 1 Pp ' '
A mapping X : M -» such that p -* X(p) e Mp is called a vector field if in each coordinate neighbourhood U the expression
/ \ i / 1 n\ °° i X(pJ a v (x , x defines C functions v on a neighbourhood of Euclidean n-space En. We denote by 3C(M) the space of all vector fields on M over the ring F(M). Similarly we define a differential 1-form as a function: a> : M -» M* . P
DEF. 1: Tangent Bundle (TM, TI, M)-: The tangent bundle of M is the
2n-dimensional fibre bundle TM => i_J M^ ; it has in a natural way pG M 00 X a G structure in the following -way: if v a v e. at p e M with local coordinates x1, .xn then we take the coordinate system
on the neighbourhood ' M . p e M P Clearly a vector field on M is simply a cross-section of this bundle.
DEF. 2: Frame Bundle (FM, %, M): A frame at p e M is any basis of the vector space Mp. The set of all frames at all points of M is called the frame bundle of M. Let e, . .... e be a moving frame on U. then (e,) ..... (e ) is one basis of M . The most general basis of M stems from this one by applying an arbitrary non-singular transformation B a (b^ ) to it; we get
f = (f,, .... f ) where f. = b.D (e.) 3-
Clearly the independent variables {x1, x1, J coordinatise
the neighbourhood of FM, and dim FM =» n + n2.
§ 2. Multivectors and Forms.
From the space Mp we can form the space of q-vectors at
p e M. Similarly we can form the space A^M* of q-forms at p e M.
By considering the union of these respective spaces over all of M,
each of these spaces gives rise to a corresponding bundle on which
00
the General Linear group acts. The C cross-sections of these
bundles are called fields of q-vectors and q-forms respectively.
Note that F(M) acts on both of the vector spaces 3£^(M) and
3£ *(M) of all q-vector fields and q-form fields respectively. Whence Si
with F(M) as coefficients we have SG^ = A^M over F(M) and5C* = A^r* .
Note also that 3£*(M) is just F(M).
We now form the tensor product:
thus forming the space of all p-vectors with q-form coefficients.
We can consider this space as the space of cross-sections of the
bundle of all elements of
q p (A M*) w ® (A M ) m G M m m
this tensor product being taken over TR . Again the ring F(M) acts
as a coefficient ring for 3£P . If we form the algebras = ^ ©3f* , X = ^ © X p over
F(M) with the usual Grassmann products, we have defined, by linearity, an operation:
r p+r Xx P x J ¥ -» V q -' s * q+s and
("J ® v) A (0 ® (p) m (OJ A 0) ® (vA <|> e 3fr v 6 £P The operation is associative and distributive and satisfies: V P r (_l)Pr+qs a)0v v e 3f en e X • 4. S § 3. The Cartan Calculus Let {ex, e^} be a moving frame (i.e. a cross-section of FM) on a coordinate neighbourhood U of M, and let {cu1, a/1} be the dual co-frame. We now form the identity transformation on and denote it as usual bydP: dP = o)1@e^ l^i^n. Clearly dP e , is intrinsic, and we shall in future omit the symbol ® . We can further facilitate calculation if we agree to use matrix notation. Let CJD be the 1 X n matrix (cu1, a/1) \T and e the n X 1 matrix (ex, e ) and further agree to T identify a lxl matrix with its single element (where superscript denotes transpose). Then the above equation becomes: 5. dP = cue (1.1) We define the exterior derivative operation d to be a mapping: X* -* IE*,-, characterised by: i q+i a) Linearity. b) dd = 0. c) On a local coordinate neighbourhood (U, x1, ...,xn) we have: df . df = —r dx fa function on U. dx1 d) d(to 0) = dco 0 + (-l)q to AG co G v* A A A * q DEF. An affine connexion on M is simply an operator d : -» 3f ^ satisfying: i) d(v + co) = dv + dco ii) d(fv) = df v + f dv. It can be proved from this that if d is an affine connexion on Mn then d induces a unique collection of operators, which we.shall also simply denote by d, one per space -» 3C satisfying: a) d(v + to) = dv + dco v,to e 3£ ^ b) d(v to) = dv co + (-1)^- v A dco v e 3£P to e 3£r X A A ' A •*» q s c) d coincides with the affine connexion on an derivative on 9£0 = £ * q * q . 6. For a proof of this result see Flanders [18J . 6 Now each e^ e 3£ ^ hence de^ 3C ^ } we can thus write de. = cap e. or in matrix form: i i J de B fie where SI = (CD^ ) (1.2) ft is called the connexion matrix and its elements (l-forms) the connexion forms. In a sense equations (l.l) and (1.2) and the operator d contain the entire calculus of an affinely connected n-manifold. For we now simply apply d to these equations and to the resulting equations. We define the torsion matrix T of M11 to be the 1 x n matrix of 2-forms: d(dP) = Te where x = dco - ojfi (1.3) Its elements are called torsion forms and clearly Te e . Equation (1.3) is the first Cartan structural equation. We define the curvature matrix 0 of to be the n x n matrix of 2-forms: d(de) = 8e where 0 » dfi - ft2 (l.h) Clearly 0e e 3£ * and we call the elements of 0 curvature forms. Equation (lA) is the second Cartan structural equation. Applying d to (1.3) gives: 7. dT + TQ = cu© and from (l.*0 we get the Bianchi identity: d@ = CIQ - 6fi Clearly T a 0 (" to a coft I0) 0 = 0 Flanders has shown that the following identities hold: 1) d0r = fi0r - e^fi 2) d(Trace 6r) = 0 r 3) d(o)0r) = T9 + o)0rn k) d(T0R) = o)0r+1 - TGFfi 5) d2r_1P = otf^e 6) d2r_1e = fi0r-1e d P = T0R e r - 1,2... d e - 9 e . The above system of equations is closed under d and expresses every possible result of iterated application of d to any of the basic quantities. Let v 5= Xe be a vector field (i.e. v e 9E1 ), where X is the o 1 X n matrix of functions (X1, Xn). Then d2r_1v = (DX)0r_1e 1 where DX = (DX1, ..., DXn) d2rv = X0re r = 1,2.... J = dX + Xfl and we have d(DX) = DXfi + X6 . 8. § h. Change of frame and global forms In order to derive the local and global transformation equations of the above matrices let e , e be two moving frames defined on the same local coordinate neighbourhood U. Then e = Ae A a non-singular matrix defined on U. From dP a me = me we have cB = oA""1 From Te = re follows T = TA"1 . Differentiation of the first equation yields ft = A ftA" 1 + dAA*"1 (1.5) Similarly 6 = A e A-1 -1 and from A = \A we have D X = D X A*"1 Equation (1.5) is basic to the theory; it tells us how the various matrices ft associated with the various moving frames e must be related in the intersection of two neighbourhoods if we are to define an affine connexion on MN globally. Consider FM(DEF 2) the frame bundle, and define the forms ~i ~n _1 3S = o)B where 3S = (S51, ...,03N) 9. These are 1-forms on the part of FM over U; the values of the at a point f £ FM given by f. = bJ (e ) are x S5|f = (o>\v)B~ . Note the 2S1 are independent of any connexion. If now U is a second coordinate neighbourhood, e a moving frame on it and p e U fl U then: e = Ae and a: = a>A-1 . If f e FM has coordinatesB w.r.t. e and B w.r.t. e then Be = Be = BAe Thus 25 = OJB"1 = (SSA) (BA)"1 = a>B_1 . This implies that the (linearly independent) 1-forms S31 are globally defined on FM and independent of the particular local coordinate neighbourhoods and moving frames used in their definition. In a ~ i similar way we can construct global connexion forms :- 7t = (u^ ) = BOB"1 + (dB)B-1 and hence finally we define the general torsion and general curvature forms respectively: T' = dcS - cuft § = (6? ) = dfi - ft2 . 1 ' 10. Note: The classical formulation of the above can be obtained by expressing all forms in terms of the basic forms co1. Thus 1) (Bp = r.J' ojk 2) 26? = R.j,,coku/ i lit x 1 K.* 3) 2T1 = T1., J cok k) DX1 = X1. (j define the connexion coefficients, riemann tensor, torsion tensor and covariant derivative respectively. DEF. 4. A Riemannian manifold is the structure consisting of a C manifold M together with an inner product on each tangent space such that whenever v, cu are vector fields then their inner product oo n is a C function on M . One of the salient features of a riemannian manifold is the existence of a unique symmetric affine connexion d such that: d(v.co) = dv.u) + v. doj § 5- Submanifolds Let Mn be an immersed submanifold of a riemannian manifold n+ M ^ } with the induced metric. We shall work with the bundle of adapted frames: let {e^} be an orthonormal frame at p e M11 such n that {ex, e"nJ are tangent to M and ^n+\> •'•> ^n+jf^ normal to Mn. We shall agree from now on to the following index convention: 11. A,B,C = 1,2, .n, n+1, n+N i,j,k = 1,2, n. (1.6) oc,p,7 = n+1, n+N From § 3 e dP = cue + ffie where e = (en+1, n+N) (1-7) de = He + fie -T = [ ^ de = -fi e + fie -> where o3 = (a/1*"1, a>n+N), fi = (a^ ), ?S = (a\°), fi = (u>J). Also d(dP) = Te + re (1.9) B d(deA) = eA e]E 5B For a riemannian manifold torsion matrix = 0. Hence -A -B - A dCD = CD Oig and (1.10) __B - C - B ^B dcoA = o>A coc + 9A The equations (1.7) through (1.10) contain the geometry of Mn+N Now - _ - gAB = eA'eB + g = = 0 —* '"A gCB CA ^AB _ B _,_ A . . _ _ ffl + = lf S A "B ° AB = BAp i-Lo 12. Wow onM11 o> = 0 (1.11) da) = o& and CDH = 0 . (1.12) The equations (1.7) through (1.10) together with (l.ll) contain the geometry of Mn . X e e DEF. 5. Normal Bundle (TM , TI, M): Let (e.^...^ , n+iJ* ••^ n+jjf) be an adapted frame then (e , e 111T)- is a normal frame n+1' ' n+N (i.e. an orthonormal basis for the normal space M^). The set "LJ M is called the normal bundle of M with structural group 0(N) peM P N acting on the standard fibre HR . DEF. 6. Adapted Frame Bundle: (AFM, n, M): IS the union of all adapted frames at all points of M, and is a principal bundle with group 0(n) X 0(N). It is clearly a sub bundle of the bundle of orthonormal frames over Mn+-^ restricted to Mn . If we write d. e. for d e., then we have the equation of j i e. x> Gauss: d.e. = cb.^feOe, + cu.a(e.)e (1-13) The (induced) connexon on M is by definition the tangential component of this: d.e. = tang (d.e.) = (ii^'(e.)e. j l 3 i l j k i,e' de = fie . 13- Weingartens equation is just: 1 P 1 d.e = CD (e.)e. + co (e.)eQ (LI *-) Whence the connexion on the normal bundle d - the so-called 'normal connexion* is just the normal component of (1.1*0: d.e = Nor(d e ) = u>^(e.)eQ j a j ° a J 0 i,e* de = He- J Note r a:. (ek) = (e. ) torsion zero = > , U).a (e . ) = ai .a (e. ) It follows from (1.12) and a lemma of E. Cartan that cna = A°.J (1.15) where ACT = (A°. . ) is an n X n matrix. The quadratic differential forms: -dP.de = A . .CD ® o>d are the so-called second fundamental forms of M. The curvature matrix Q = (&? ) of the induced connexion on M is X I given by: lh. & = a.a - n2 and this is related to the curvature matrix of Mn+^ by the Gauss curvature equation: m _ J 8 = 4> - a SI where 0 = (0.J ) . 1 The Codazzi-Mainardi equation is: d(nT ) = siTsi + nnT + $T where $ = (0.° ) which can also be written: dfi = fin + SISl + $ The connexion (matrix) of the normal bundle has curvature matrix 8 =(ep) © = dfi - n' and this is related to the curvature matrix of Mn+^ by •*•1 P - P i P 9 = 0 + CD CO. a a a i The Bianchi identity in § 3 becomes: dO = fl* - *fl + $fiT - fl$T d$ = Sl$ - $fi + fi$ - Ofi where $ = (Er ) d$ = $Tfi - fiT0 + fl$ - $fi 15. Note for an orthonormal basis we can write CD^ = u^g , and for a space of constant riemannian (sectional) curvature K: a „ A B = AB A °* • § 6. Some Definitions. We end this chapter with some basic definitions that we will have recourse to later on. (a) One-parameter transformation group: on a C n-manifold M11 is a set I*^.} t e 1 of diffeomorphisms of M onto itself such that the mapping $ : 3R X M -» M defined by $(t,p) = ^(p) P e M, satisfies: (i) $ is differentiable (ii) *(s, *(t,p)) = *(s+t, p), i.e. 4>s+t = 4>s *t VB,t (iii) 3>(0, p) = p i.e. * = identity. Such a group defines a tangent vector field X on M in the following way. Let f e F(M) and set: f f(*t(p)) - f(p) 1 d b x. (f) = Lt J — - y = ~ f(*,(P)) P t->0 I t J tit Z ^ 16. We can think of this field as follows: at every p e M, the mapping t -» The following theorem is a partial converse of the above: Theorem: Let M11 be a smooth manifold, X a vector field on M, U an open set in M with compact closure K. Then we can find £ > 0, and for each t with |t| < e, a map *^ of U C M such that (i) Map *:UXE-»MX]R (E is the set |t| < e) is a diffeomorphism onto an open submanifold. (ii) If | s|, |t| and | s+ t| are < e; p and "^(p) are in U, then: +g-*t(p) - *B+t(p) '• (iii) For each p e U, f e F(M), xp(f) - ^f(*t(p))t =o _ The map 0 is completely determined by these conditions. Corollary: If Mn is compact, each vector field generates a 1-parameter group of diffeomorphisms of M. DEF. 7- A vector field on M is called complete if it generates a 1-parameter group of diffeomorphisms of M. Note that in the product 17. M X 3R the field cVdt, which maps to zero on the first factor and to the standard field on the second, is complete, here •t(X,s) = (X, s+t ) This serves as partial justification of the term 'tangent* vector since clearly such vectors correspond to displacement along M. (b) Lie derivative £ DEF. 8. The Lie derivative of a tensor T of type (r,s) with respect to a vector field X is defined: (£T)(0j_, 0r,X1, Xg) = X(T(01, 0r,X1, Xg) -T(£01,92, X e .x s> -T(e • • •} e ,x ,• • • > x ^[x,x ]) r' i' s-1' s where 6 ..., 0 and X • • •} X are respectively arbitrary r 1-forms and vector fields on M; and where: (i) (£fl)(T) = X(0(Y)) - 9([X,Y]) X (ii) £f = Xf f e F(M) X (iii) £Y = [X,Yj. In local coordinates X = X1 d/dx1 and Y = Y1 X dxJ then X X x 18. Clearly for a symmetric connexion d : £Y = d^.Y - cl^X . Theorem: If X is a vector field induced by a 1-parameter group of transformations l*^.} then a tensor field T is invariant under X (or equivalently under 4>. for each t) iff £T = 0. Note that the Lie derivative is linear over sums and commutes with contraction; it is also a derivation in that for a product or a transvection of two quantities ¥ we have the Leibnitz rule: £(*¥) = (£ + 0(£ ¥) . X XX DEF. 9: Affine Mapping: Let M, M be manifolds with linear 00 — connexions. We call the C map induced map horizontal curve, i.e. if each curve T of M into a parallel vector field along curve *(T). An affine transformation is a diffeomorphism of M onto itself which is an affine mapping. Geodesies are preserved and the arc length s receives an affine transformation: s -» as + b a }4 0, b constants. DEF. 10. Conformal transformation: Is a diffeomorphism * from M onto itself such that the induced tensor g* = **g is another riemannian metric. Note that 0^ preserves orthogonality and dilates uniformly. 19. Theorem: A transformation on a C riemannian manifold (M^g) is 00 3 a positive real-valued C function f on M such that: g* = f g or g* = exp(20)g where 0 e C°° : M -» H . A conformal vector field is one that generates a conformal 1-parameter transformation group. 00 DEF. 11. Projective Transformation: Is a C homeomorphism of M onto itself leaving geodesies invariant, the affine character of parameter s not necessarily being preserved. A transformation * on M is projective iff 3 a covector pC^) depending on If X is a vector field on (M^g) which induces a 1-parameter group of transformations on M. Then in order that X define: (i) an isometry (motion) is: £g. . = X. . + X. . = 0 the comma denoting covariant derivative. (ii) an affine collination is: = ^ i k + h\l X* = ° 20. (iii) a projective motion is: £r.x = 0 X J K where (iv) a homothetic transformation is: £e. . = 2cg. . c = constant x 10 ij (v) a conformal transformation is: X 1J 1J Clearly for c = 0 in (iv) we have (i); we thus call (iv), such that c 4 0, a proper homothety. Note also that from the definition r + f A • *^'«|«lI>lk (|«*Js3-(f thus (i) and (iv) are examples of (ii). Proofs of these results are to be found in Yano's book ,The Theory of Lie Derivatives and its Applications'. (c) Generalised vector product Considerable use will be made of the vector product due to Hsiung [6], Namely, let (M",g) be a riemannian manifold (n ^ 3)j 21. and consider a fixed orthonorraal frame {p, e±} .e } at p e M. Let a_^ (l ^ i ^ n-l) be n-1 vectors at a point in Mn whose contra- variant components w.r.t. frame p^e^ e^ are a^ 1 € a n. We define the vector product of the n-l vectorsa^ to be a vector n a in M denoted X ... X n_p whose contravariant components are given by: g a g a g& a, al l a2 l an 1 g a g a n al 2 ' tr2 2 ' g a n-l an~2 El, X • « • X B J - (-D i n-l a a & >alan-l> ga2an-lJ * ffn n-l From the definition of scalar product of any two vectors a^ and a 8 viz. a. . a. = g Q a. a. it follows that a. x ... x a . is i J a0 i J 1 n-l orthogonal to a. 1 =S i ^ n-l . Clearly if n is a permutation of the set {l,2, n} then: a /lX x ... x a / .. \ = (sign Tt)a, x ... x a , ti[l) 7t(n-lJ i n-l where sign it is +1 or -1 as it is even or odd. If (a?) are differentiable functions of the n-l variables then from the definition: 22. a Xa x x a ~~~~7( i X ... X a-n_^) = y (a^ X ... X a. ,X i i+l ••• n_i^ dx J~ 5x J In a determinant whose columns are the components of vectors or vector-valued differential forms, we make the convention that in the expansion of the determinant the multiplication of differential forms is in the sense of exterior multiplication. A similar convention will be observed for vector products involving differential vectors. For the differential vectors dP. = oj.^e. 1 < i < n we i i 0 defin k, k i n i ||dPx, dPj| :=o k A ... A a) n 1 i' ' n' l n where | | means a determinant of order n whose columns are components of the respective vectors ex, e^ . (d) Newton's Formula Let A° be matrix of second fundamental form in normal direction e^ (see (1.15)). Then we define the principal curvatures X^ (cr) 1 $ i ^ n at a point p e M to be the roots of the equation: |Aa - XG| = 0 G = (g...) and consider the k^*1 elementary symmetric function of the X_^(a) : ) X. (a) ... X. (a) S SR := ^ ••• \ (°) o i,<.. . n Set Y (X.(CT)) i=l Then the relations between S, and P are given by the formulas of Newton: Pi " Sx = 0 P2 - S1P1 + 2S2 = 0 P S P + S P 3 " x 2 2 i " 3S2 = 0 P + S P + + n 1 S 0 n " Vn-1 2 n-2 " - ("^n-!*! ^ ^ n = ' We can thus express Sfc in terms of P1, PFC as: S. = Pi 2 21 S2 = -P2 + Px a 52 S3 = 2P3 - 3P1P2 + Px 2 h'. s4 = -6P4 + SPgP-L - fiP^P., + 3P2 + tj+...+tn+k t s, = P N n t1+2t_+...+nt =k Cti*. )(t2i).-.(t I )2 12 n t. > 0 l + tx+..• tn+n (-D X 1 t^t^+.-.+nt =n^i!)(t2:)*--(tn^2 t. ^ 0 1 2k. On the other hand we have the identity: SS, - (k + l)S, . = n i k v ' k+1 \ V (X. (a) - X (a))2X (a)...X (a) " ^-J .1. X2 13. -""k+l ix < i2 i3<-" DEF. 12: For the normal direction e we define the k**1 mean a curvature M^(a) to be: n where k; \ = S\, wher e v( k ) is the binomial coefficient. Whence: _ >. (n-k-l)I 2 M k X. (a)) X. ... X. 'A " \+l = V ^ Vl i3<... CHAPTER II We now develop the local curvature theory of a riemannian n-manifold immersed in Euclidean (n+N)-space; the results however are easily translated without any essential change to submanifolds of a space of constant curvature. We first develop the more classical theory with more recent extensions due to Chern [11], Hsiung [7], Gulbinat [13] and Leichtweiss [20]. § 1. Standard Equations JI n+K Consider an immersion x : M -» E , we denote the position vector of x(M) relative to a fixed origin 0 by x. If (u1, u11) are local coordinates on M then the (induced) metric onx(M) is given by: g. . = < x.,x. > x. = dx/chi1 a As before we define the outer product [a2, °f the vectors the (oriented) Euclidean space En+^ by defining a a a a < a, [a2, n+jj] > = \ ) 2> •'•> n+jj-l Va . We choose the orientation of the orthonormal basis of the normal bundle so that: xn, en+1, en+N| >0 26. Clearly 1 i, i n+l n+N1 in in and r A 1 _ /_, \n+oc-l i, ' i ' n+l a n+N 1 a in n • where as usual the carat A denotes omission of that element, and where the Ricci e-tensor on Mn is: G = g2sign(i,, i ) where g = det(g ) in. • and hence 1i* • --. . e . = kl ) g. \..-g. /. \(sign TI) L 1 u N u x ^••^n Jk+1 °n • ~ k+l k+1' n n - (2.1) summed over all permutations u leaving (i^ i^) fixed. We also have: A r t _ ci . l ' l, i n+l ' n+N i . ..l,...! l l k n l k n corresponding to each normal vector e^ we have defined the symmetric second fundamental form tensor: ACT. . • - < x., d.e > (2.2) d.e = -g° A., x. + a) (x.)eQ iff B it j a ^ i' P 27. where a) v (x.) = < d. e , eQ > and a> v (x.) + u> vQ (x.) = 0 . a i' I cr P a i' P i' The conditions of integrability are: likj " ik " ij k£ ff a P a P A ., -A° . +a>R (xjA . -a3ft (x.)A v = 0 10 ,k ik;,j P V IJ p j7 lk (2.3) where R. . is the curvature tensor. DEF. 13. Mean normal curvature vector i) : is defined by: ^ Trace A° = g(ea, TI) V e a whence: Ti = ( - ^ (Trace A°)e \ n J c § 2. Generalised classical curvature theory We now define generalised second fundamental forms which enable us to define generalised mean curvatures. Let e^ be normalised i. cross-sections of TM } and define: 28. A _/ w \ = A. . A. . ...A. . I(r)J(r) ixDi r ^ r and A°_/ /(r) = A°.j UB. A0. 'r I(r) - ix i2 ir where l(r) := i^} , J(r) := j^} are ordered sets of numbers. We can now define a tensor symmetric in (ij) as follows: i i 1 i ,:L J a id i'" s, s +2" n ^i""^, k k ii!C = e Ls, +2" 0l °k X A A Kr^jCrJ I(rk)j(rk) and where U I (r, ) U {i ..., i } = {1, n} t=l k and I(rg) n I(rt) = * * t . And clearly there exists an analogous relation to (2.h) for the r sets of indices j(rj_), J( k) • From (2.^) we define the generalised mean curvature of the sub- manifold to be: CTk \ / CTi »k^ H k ^ = C 29. Clearly the curvatures depend on the choice of the normal vectors e . In case k = 1 the curvature functions H(° ) are a r simply the usual curvature functions with respect to the normals eff , namely the r^ (normalised) elementary symmetric function of the principal curvatures X^(a). We note that nC ^ 0 ^ = G"1 and H ^ ^ ^ = 1 where G = (g^) . If e and eQ are two orthonormal vectors then e (X p where a p is a unit normal vector, and we have: Lemma 1 (Gulbinat): The rtn curvatures H(^), H( ^ ) and H( ^ ) are related to the unit normals e^, e^ and e by the relation: r-1 <^:)-»(:M:)-X(;)»(: s-t t=i From (2.j) we have the well-known R = AaG-1Aff - nH ^ ACT (where R = (R„)) (2.5) Whence for the scalar curvature K we have: n(n-l)K = -Trace R = . . . 30. Hence from the first equation of (2.3) we have the well known generalisation of Gauss' theoremaegregium: K = H Theorem 1: On x(M) , ^ ( 2 a the right hand side depends only on the metric and not on the normal space. Proof Transvect the first equation of (2.3) with iki3...i rj e e 13"'1n and use the properties of the outer product defined earlier, together with the fact that: iki„...i e 3 nA° = 0 lk recalling that e is skew symmetric. Lemma 2: Let ea be an arbitrary normal to X(M), then it follows: (a) n(n-l)c(^ = nH( « )g^ - gik J* . a iJ ij (b) n(n-l)(n-2)G(2 ) = n(n-l)H(« )g + 2^V/A^ 31. Proof: From definition of tensor C n ii ikiQ...i 3 3 n T n!C(?) = e e* . . A°v 1 i„...3 1 kr n / 0\,/ ij kr ir kj x a (n-2): (g g - g g )A kr (b) is proved analogously using the fact that from (2.5) we get J n(n-l)H( 2 ) = n-TTC x ) - A g g A kf . Mote finally that we have: 2C(1) g A f = H(2)g -(n-2)C(g) whence by (2.U) and (2.5) P 2 tt P a ki n(n-l) H(J| 1) = n H(1 )H( 1) - A .. g ^ A\£ These two equations together with (2.h) have been used extensively in investigations in minimal surfaces and umbilic sub- manifolds. 32. § 3» Canonical curvature theory We now develop a curvature theory of the mean normal curvature vector. In order to do this we generalise the classical second and third fundamental forms of surface theory. DEF. lk. The second fundamental form of X(M) is defined to be the quadratic form: A. . du1 duJ = DEF. 15. The third fundamental form of x(M) is defined to be the quadratic form: E. . du1 du^ m < x. . . x . > grk du1 dud . These forms are clearly independent of the choice of special sections of the tangent respectively normal bundle. In the case of co- dimension 1 (i.e. hypersurfaces) this third fundamental form coincides with the classical one but this second fundamental form is distinguished from the classical one A du1 du'' bv a factor: A = nH(l)A A =« (A. .) A = (A. .) where H(l) is the mean curvature of the hypersurf ace. Note: The above generalisation of the second fundamental form is 33. meaningful in the sense that it is in general impossible to normalise T] globally since there may be points where T\ = 0. In case TJ J£ 0 on x(M) then we can normalise it and hence: A. . = - < d. T]. x. > thus A = nH(")AA EiJ - A ik 8 A fj • DEF. 16. We define the principal curvatures of x(M) to be the n roots X. 1 ^ i < n of: I |A - XG| =o and we further define the (normalised) elementary symmetric functions of the principal curvatures; in particular we call: 1 1 i d i H = H. = - Trace A = - g A. . 1 n n the (first) mean curvature of x(M). From this: nH = A submanifold for which H = 0 identically globally is called a minimal submanifold. We can now derive the derivative equations and the conditions of integrability (i.e. (2.5)) for t] . Let xQ be any point not in x(M). We agree to identify points n+N in E with their position vectors in the usual manner. Then 1 1 x - xQ = (i)g " d±(< x-xQ, x-xQ >)x.. + < x-xQ, ea > ea . Now from: n = nH(1)ea . a . and < x-xQ, T] > » nH( ^ ) < x-x0, ea > we get the derivative equations: k;j P a d. il - -g A.k x. + n{d.H( J ) + o>a (x.)H( i ) } ep . If we denote the term: n{ } by dP^ (= < d^T], e^ > ) we get for the conditions of integrability: A. . . - A.. . = AP. . dft - AP.. dp. ij,k ik,j 10 k ik j ^i,k - <,i + ** \i ^i ~ ** AU A^k = P = d\ % (x.) - da. . In what follows we use the: 35. Theorem: The mean curvature of a submanifold for any normal orthogonal to the mean normal curvature vector is zero. In case H 4 0 on a neighbourhood U of a point then T} is normalisable in U. Let n = + < TJ, TI > 2 > 0 then nH( J ) = - > 0 . x 1 n Further let e , n+1 < a =£ n+N, be an orthonormal basis for the cr ' normal space such that ^n+^ is in the direction of TJ. Then l) A(l) = |JA where A(l) is the second fundamental form in direction e 'n+1 2) H( J ) = nH 3) < x-xD, en+1 > • |i < x-xQ, T) > . We now discuss the relationship of these generalised funda• mental forms and curvatures derived from them to the intrinsic geometry of x(M). We first generalise the classical relation between the three fundamental forms of a surface in E3. Theorem: On x(M) R = E - A E = (E. .) ij £k a Proof: From R. . = A^ g A .£ - nH( « ) A\ . and DEF. l'+ and 15. 36. using n(n-l)K = -Trace R and theorem 1 gives R + (n-l)KG = E - A + (n-l) ^ H( £ )G Q.E.D. Wow every riemannian 2-manifold is an Einstein space, thus the left hand side vanishes in this case. Whence using A = nH(l)A we get the classical relation: E - 2H(1)A + H(2)G = 0 . We have thus proved: 2+HT Theorem: On any 2-manifold in E we have E - A + KG = 0. Theorem: The following three tensors are intrinsic: (a) E - A + (n-l)KG n 2 Cb) iJ ik a £ C(«) g A .3 * (c) Y C(«)±: n > 2 a 37. Proof From Lemma 2. n(n-l)c( " )^ = nH( " )giJ - g1* gJk A*^ hence ^ S^ BlJ and 13 1 k nCn-D^nHf^Cf^) - nHg« - g ' g^ A{k and for n > 2, again from Lemma 2 iJ £i n n(n-l)(n-2)G(«) = n(n-l)H( « )g^ + 2g A^ g^ g^ A^ i£ k a -2nH(«) g g^ A £k that 1J ij ik n(n-l)(n-2)^C(^) = n(n-l)^H(^) g + 2g (Efcf - A^) a = n(n-l)K g1J + 2R1D whence (n-l)(n-2)(n^C(^)1J - Kg^) = 2^ + (n-l)KgiJ) ik = 2g g^ (Ek£-^+ (n-l)Kgkf) Q.E.D. 38. We note finally that it is possible to generalise the tensors (b) and (c) of this theorem as follows: C(k)1^ = ) C(£)1J 0 k ^ n 1J *W . Vc( ^ g« A« , a However in this generality little if anything appears to be known. We end this chapter with a theorem which we shall use implicitly throughout the remainder of our work. Theorem: Let J^be a riemannian n-manifold and e,, ..., e be an orthonormal basis for the tangent space on a neighbourhood U of M, then d e E 0 ( i A ••• A N) " • Proof: From (1.2) de. = a>.J e. I i j and since d < e., e. > = < de., e. > + < e., de. > = 0 i 3 13 13 thus ^-k ~ , , k j i_ _ < ax e, ,e.>+ Hence n d(en ... e ) = > e, ... de. ...A e v 1 1 A An Z_i A An i=l n e = \ i A---A Vl A^^k A A..-A en i=l = (I^1 )8A""l 6An = ° i Q.E.D. ho. CHAPTER III In this chapter we derive our main integral formulae for a submanifold of a riemannian manifold admitting a complete vector field. We shall then derive the formulae in case of some special vector fields. § 1. The generalised Minkowski formula Lemma: Let (M,g,d) he an n-dimensional riemannian manifold with a connexion d, and let (U, u1, un) be an allowable coordinate chart on M with local coordinate field e^ = b/bu Then the Lie derivative of g w.r.t. a vector field X = X1e. 1 on U is given by: (i) (fig),, = (fig)(e,,e ) = < d X, e > + < e , d X > X ^ X ^ 1 0 1 J or (ii) (fig).. = d.X. + d.X. . X 1J 1 J 01 Proof: (i) From the definition of £ and the fact that M is riemannian d^Y - d^X = [X,Y] = fi^Y and cLg = 0 equivalently cL< Y,Z > = < dY, Z > + < Y, cLZ > in. we have: (£g)(e. e ) = X < e , e > - <[x,e ], e > - < e , [X,e ]> (3.1) = < dxei - [X,e.], e.. > + < e±, ^e. - [X,e.]> = < d.X , e. > + < e., d.X > . (ii) d.e. = cjo.^fe.Je, cL,e. = X^cu^fe.Je, Also [X,e.] = - e.(Xa)e.. . So in local coordinates (3.1) can be written (£g) = (£g)(e,,e,) = (e (X*) + X* w ,k(e.) )g, + (e (Xk) Vi ^ + ^ik dd d.X. + d.X. Q.E.D. Let f : M -» M be an isometric embedding of the manifold (Mn,g) into the C °° manifold (Mn+^, g). In terms of local coordinates (u1, un) on a neighbourhood of a point p e M and h2. local coordinates (x1, xn+N) on a neighbourhood of f(p) e M we have A x = Au1, un) (3.1A) where at each point -\ A A A rank (x. ) = n where x. = —r au1 and y, . = < r>_ R = (p. ) 1J ou1 8 AB Let M admit a one-parameter group T of transformations generated by a vector field £, the transformation being given by: x = x + £ dt . We assume that the submanifold lies in a domain of M which is simply covered by the orbits of the transformations generated by the vector field £ . The field | is assumed of class* C°° and non- vanishing on Mn, and at no point of Mn does it lie entirely in the tangent space to M11. In this section we work with the adapted frame bundle AFM (DEF.6) n n {e^, e0} , e^ being tangent to M and eff normal to M . Let e(£) be * Actually Cr (r 5 3) is sufficient. *3. a unit vector normal to M that lies in the (n+l)-dimensional vector space spanned by the linearly independent vectors: [e^, .. •, e^, £} . We can choose a basis of the normal space to Mn so that the vector is included in it, and we will assume this basis ordered so that: e n+^ = e(|), i.e. e(£) is the first normal vector. At each point of M*1 we consider the following differential (n-l)-form A , : n-1 Vl := He^>en+2' en+N'6' ^ (3"2) 1 n^T ^ which is by definition: ix in-1 n = |e(i),e 2, e , g, e , e |u> A . .. A a> i n-1 where dP is the identity transformation on the tangent space at each point. Note that this form is independent of any origin and depends only on i and M11 . Whence: ^n-l = dHVl'V2> W S,dP' dP" hh. + "en+l>d%+2> "•> en+N' ~ > ^1 + l|en+l' en+N, d£, dP, dP|| (3.5) We note that since Mn is riemannian: d(dP) « 0 and dv ss v . co1 or Dv1 = v1 . Now de_ = OLI1 e. CT a 1 CT = -A .. J e. (3-3A) ij 1 i.e. e. . d.e,_ = -A . . Hence where H(* ) is the (first) mean curvature in direction e(£) = en+^ P(£) = < en+v | > n 1 1 *1 = volume element on M = ou A ... (a . h5. Similarly for each normal vector e_ , n+2 =S a ^ n+W we have: "en+l' iea> '"' en+N' ^ dP; '"' ^ Now by definition the vector £ lies in the (n+l)-dimensional vector space spanned by: U-L, eR, e(£)} Hence the right hand side of (3»M is zero. Thus l|e(i), en+2, dea, e^, £, dP, dP|| = 0 n+1 < a < n+N On the other hand, from the lemma, the last term on the right hand side of (3«3) becomes: l|e(6),en+2, en+N, d|, dP, dP|| „ (n-l)f.(i)(-l)N^N+n_1^ Trace f*(£g) g £ where Trace f*(£g) = gld (f*( £ g)). . g • £ I 1J Collecting together we have: = ^jNfN+n-l) |H(|)p(|)*l+~ Trace f*.( £ g) | Integration over now gives: 3M W11 M" where is the boundary of Mn. Now in case is closed (i.e. compact and without boundary) we use Stokes' theorem to get: — 00 . Theorem A: Let M be a C riemannian (n+N)-manifold admitting a complete vector field I, and Mn a closed n-submanifold lying in a domain which is simply covered by the orbits of 6. If M11 does not contain any singular points of £ and the set of points of M where £ lies entirely tangent to M is null; then: jV *)p(|)* 1 + ^ I"Trace f*(£g)*l = 0 (3-5) M11 Mn g 1 This integral formula generalises to arbitrary dimensional sub- hi. manifolds in riemannian spaces of arbitrary co-dimension an integral formula obtained by Chern [n] in 1952. We may interpret A^_^ in the following way: consider the tangential component of | tang £ = e . Now take its dual 1-form and denote it by K Kv = s5i ODi If we now apply the Hodge star operator to this we get: 0 0 00 *K = ) 6 (-1) a A A • • • A which is precisely A-nm_± • § 2. The second generalised integral formula -n+N , We first specialise the ambient manifold M to be one of constant riemannian (sectional) curvature. On the submanifold M11 we consider the following differential (n-l)-form B(r)n_^ : B(r)n-1:= l|e(l),en+2,...,en+N, g, de(|),..., de(|), dP^.^,dP|| r n-r-1 where 1 =£ r n-1 Whence to each fixed r, there corresponds a differential (n-l)-form B(r) . . n-1 h8. From the classical Codazzi-Mainardi equation it follows that on the submanifold M we have: 1 k 1 Pi o ! -a) ai — to ax, = 0 a OK a p a (3-6) ,_T, _T =_T _T i.e. 6.(0. ) = !l!l-fi!l + $ where 6 is the curvature tensor of M. Now M has by assumption constant riemannian curvature, hence: d(de(£)) = 0 . (3.7) Hence by applying the operator d to B(r)^ ^ we have: (ffi r d e ( )n_l = l (0,en+2,...:,en+N,|,de(i),...,de(0,dP,...,dP|| II de(6),en+2,...,en+N,6,cle(£),...,de(6),dPJ...,dP|| + |[e(g),den+2,...,en+N,6,de(0,...,de(0,dP,...,dP| + ||e(i),en+2,...,den+N,|,de(i),...,de(|),dP,...,dP| + l|e(l), en+2'""'en+N'dkde^'"*'de^' dP,...,dP| + lle(l), ... + *9. + |e(S),en+2J...,en+N,6,de(|)J...,d(de(|)),dP,...,dP| Now from (3.6) we have: |e(5),en+2,...,en+N,6,d(de(6)),...,de(|),dP,...,dP|| = 0 . And from (3.3A.) we have: IM£)>en+2,...,en+w, 6,de(6),...,de(|),dP,...,dP|| = is the r+ h n where H ^r+i^) ( l)^ mean curvature of M in the direction e(l), i.e. if k^(£) 1 *S i ^ n are the solutions of |An+1 - XG| =0 then H (^.+1^i s ^he (r+l)th elementary symmetric function of the k^(£) divided by the number of terms: (A)H(A) - 1 v« \(t,-\«ct) • ix Note: From a theorem in linear algebra it is known that the r*1*1 elementary symmetric function of the characteristic roots of matrix An * is equal to the sum of all r-rowed principal minors of An+^ . 50. H d efiried a Hence ^r+]^) (r+i) - "bove is equal to the sum of all (r+l)- -rowed principal minors of A . In particular nH( J) = Trace A1n+ 1 H(5) = detAn+1 n i=l where A11 is the cofactor of A.^+* in det An+^. 11 It follows further from (3»3A) and the definition of generalised vector product, given earlier, that the vectors: e(|) X den+2 x e X...X en+w X de(£) X...X de(g) X dP X...X dP r n-r-1 e x e de x (£) n+2 X n+5 •••X en+N X de(|) X...X de(£) X dP X... X dP e(5) x en+2 x ... x den+N X de(£) x ... X de(£) x dP X ... X dP have the same direction as e e .... e ,„ respectively. n+2' n+y ' n+N J Hence: 51. e l (0,den+2,en+5,....,en+w,6,de(|),...,de(g),dP,...,dP|| = 0 e(0,en+2,den+5,...,en+r |,de(6),---.de(g),dPJ...,dP|| = 0 ||e(g), den+N,£,de(|), de(g),dP, dP|| = 0 since the vector g lies in the (n+l)-dimensional vector space spanned by the vectors: {e1} en, e(g)} . Now the vector: e x e (0 n+2 X ••• X efi+w X de(g) X ... Xde(g) X dP X ... X dP is orthogonal to the vectors e(g). e ,_. .... e and from (3.5A) n+2 n+N' ' de(g) = 03n+^" e^> we have: l|e(8), = nK-D^-^^TracetGC6) f*(£g)]*l W r t where C(r) = (C(p ) ) ii i , g .id 1 i'-- n-l j^i'-'j- n+1 n+1 °( v ) = 7—TTT € e A. ... .A. . g. 1n-lJn-l 52, where as before e is the Ricci e-symbol of the sub- manifold Mn . Hence we have finally: ~T dB(r)n_1 = ~ d||e(6),en+2,...,en+ir, t,&e(Z),... ,&e(t,) ...,dP N(N+n-l)-r = (-D X Trace[c( ^) f*(£g)]* 1 j Integrate both sides of this equation over Mn rr l|e(E), e e 6 de n+l>-*-> n+r ' (^» ...,de(S)jdP,. ..,dP| 1 - (-l)^- ^ [H^^l +(±)/Trace[C(^)f*(£l)]*l and in case is closed we use Stokes' theorem to get: Theorem B: Let M be a C°° (n+N)-dimensional riemannian manifold of constant curvature admitting a complete vector field and a closed n-submanifold lying in a domain which is simply covered by the orbits of |. If Mn does not contain any singular points of £ and the set of points on where £ lies entirely tangent to M11 is null; then: 53. H p(5) 1 + Trace[c l f £ 1 0 / (Ji) * (|;)/ (r ) *( g)]* = M M (3.8) This is the second of our two generalised Minkowski integral formulas. The remainder of our work will involve the various specialisations of (3«5) and (3.8) and the applications which follow therefrom. § 3« Specialisations of g — CO . . Theorem 1: If M is a C riemannian (n+N)-manifold admitting a one- parameter C° group T of conformal transformations generated by a vector field g. If further M" is an embedded closed orientable C n-submanifold which is simply covered by the orbits of T, and does not contain any singular point of g. Then f H( J) p(g)*l + J* and.in case M is a riemannian manifold of constant curvature we have also: fn Ql^j p(i)*i + f *H(*)*I = o (3.10) 1 ^ r *S n-1 Proof: In case T is conformal £ satisfies: £g = £. . + I. . = 2 =i> Trace f *( £ g) = 2n * (3.11) g . £ Also Trace[C( £ ) f* (£g)] = 2nDH(5) (3-12) Substitution of (3»H) and (3.12) in (3.5) and (3.8) respectively gives the result. Q.E.D. Theorem 2: If M is a 0°° riemannian (n+N)-manifold admitting a one- parameter C° group T of homothetic transformations generated by a vector field i, and is an embedded C closed orientable n-manifold which is simply covered by the orbits of T and does not contain any singular points of £ , then: 55. fn( \) P(6) *i + CV(M) = o (3.13) where c = constant and V(M) is the total volume of V^1. In case M is of constant riemannian curvature /H(r!l)p(£)*1 + °f H(r)*1 " ° bAk) 11 1 r M M* 1 < < 1 1 r =£ n-1 . Proof: In case T is homothetic then (3.12) above. Hence: £g = 2ci Trace f* (£ g) = 2n c S I Trace [C( f*(£g)] = 2ncH( ^) r § r And again the result follows by substitution in (3«5) and (3.8). Q.E.D. Note that in this case M becomes Euclidean (n+N)-space since: _n+N Theorem: If M is a riemannian manifold of constant curvature admitting a one-parameter group T of homothetic trans• formations, then either M is E11+^ or T is isometric. 56. Theorem 2 has the following important corollary: _n+N Corollary: With the conditions as in theorem 2, if M • is taken to n+N n n+N be E and assume M contains the origin 0 of E , then if we take for £ the (nomothetic) radial vector field, where the orbits are simply the lines through 0 then £g = 2g giving for (3.5) and (3.8) respectively: f H( I )p(6)* 1 + VfM11) = 0 and ^H^J^pCO*! +^ H(^)*l = 0 1 $ r =£ n-1 . These generalise the two corresponding formulas of Hsuing [7 ] for hypersurfaces to arbitrary co-dimension. The formulas of Hsuing themselves being generalisations of two integral formulas due to Chern [H ] for surfaces in E3. _n+N oo Theorem 3' If M is a C riemannian (n+N)-manifold admitting a one-parameter C° group T of isometric transformations n °° generated by a vector field 5, and M is an embedded C closed orientable n-manifold which is simply covered by the orbits of T and does not contain any singular point of £, then: 57. f = o (3.15) In case M has constant riemannian curvature we have: f H ( 1 } p(£) *1 = 0 (3.16) J \r+V . 1 ^ r ^ n-1 Mn Proof: In case T is isometric then t = 0 in (3.11) and (3.12) whence the result follows by substitution as before. Q.E.D. As a final specialisation of the field | we can consider the so-called tension field of the embedding. This we explain briefly - for a more detailed exposition see Eells and Sampson [21] . Let (M,g) and (M,g) be complete C°° manifolds of dimension n and n+N respectively, and suppose M closed. For each point p e Mn let <, >^ denote the inner product on the space of 2-covariant tensors of M defined: P ik ik jd i < a,b >p := a bfcJ g g where g g^ = 5 and a = (a. .), b a (b. .). Let f e C°° : M -» M, then through f, g induces a metric f*g on M. We can thus define the functional on M: P -» < g(p), (f*g)(p) > • ir 58. We call e(f)(p) = ± E(f) = y e(f)*l . Then from (3.1A) Hence E(f) . * f g^f.V^-i i.e. !(f) = \ / Trace (f * g) * 1 J g where ij A B integrand = g f± f. g^ Let H(M,M) be the totality of smooth maps: M -» M. Then for each f e H(M,M) the set of C°° maps u : M -» TM such that: 7tou = f is a vector space, which we denote H(f), with algebraic operations defined point-wise. We call a typical element u e H(f) a vector 59. field along f. Space H(f) is given the inner product (denoted < , >f) defined: 1 u v e H f < u, v >f := ^ < u(p), v(p) ^(p)* > ( ) For any u e H(f) the directional derivative of E in direction u i.e. where ffc(p) = expf^ j(tv(p)) t=0 is the end point of the geodesic segment in M starting at f(p) and determined in length and direction by vector tv(p) e . We can thus define a unique field f(f) along f: VE(f) = - < T(f), u >f u e H(f) This field is called the tension field of f. Whence: -r(f) = Af + Trace f* Where co is connexion matrix on M and where A is the Laplace-Beltrami operator on M. Maps for which f(f) = 0 are called harmonic. In case M is flat 60. then T(f j = Af; in particular if M is E then a map ji" n+W f : W -» E is harmonic iff f = constant, by Hopf^s Maximum principal Let f : M -» M be a riemannian immersion, whence for each p e M, f-x-Cp) maps isometrically into j , i.e. g = f*g . We use the following theorem due to Eells and Sampson [21 ] . Theorem: Let f : M -» M be a riemannian immersion. Then e(f) = n/2 and T(f) coincides with the mean normal curvature vector. Note that for any W e M„/ > normal to M we have: < f(f), N > = Trace A(N) A(N) being the second fundamental form for N. Hence in case I is the tension field of the (isometric) embedding: < e(|), 6 > = H = (first) mean curvature of M" and where we assume implicitly that £ ^ 0 on Hence in this case (3-9) and (3.10) become respectively: J" (H2 + *) * 1 = 0 J* (H(r+1)H + (3-13)JI (3»1*0 "become respectively: J* H2 * 1 + cV(M) = 0 j" (H(r+1)H + cH(r))* 1 = 0 1 ^ r « n-1 M11 and finally (3.15), (3'l6) become respectively: J* H2 * 1 = 0 ^H(r+1)H * 1 = 0 (l^r^n-1). 62. CHAPTER IV In this chapter we use the integral formulas (3-5) and (3»8) and their various specialisations to prove some characterisation theorems. § 1. Characterisation in codimension one DEF. 17- Normal coordinates: Let 0 be a point in a riemannian / ^ -n+1 (n+1;-manifold M , and U a sufficiently small neighbourhood of 0, such that in U there exists a unique geodesic y joining 0 to every point p G U. Let y^ 1 ^ i ^ n+1, be n+1 mutually orthogonal geodesies through 0. Then the normal coordinates {y1 } of p w.r.t. are 1 the geodesic from 0, yx> ..., 7n+^ defined by: y = s cos(7,7^) where (7, y/) is the angle between y and y. at 0, and s is arc length from 0 to p along 7. Clearly £ cos2(7, 7.) = 1. If on i 1 _n+l M there exists a unique geodesic arc with minimal length joining a fixed point 0 to every point p, then we have a global normal coordinate system at 0. In what follows we shall have need of the following two lemmas, the proof of which is to be found in Hardy, Littlewood and Polya - Inequalities (pp. 52, 1C4, 105). 63. Lemma 1 (Newton): Let be the r^1 elementary symmetric function of n real non-zero numbers k., 1 ^ i $ n, and define H = 1 and (n)H := S for r = 1, .... n. Then : o r rr 2 H1H^1-H <0 l^r^n-1 r-1 r+1 r if in addition H , H H . are positive r' r-1 r-i H . / H 5= H / H .2s ... . , / H . r-1 r r-2 r-1 r-i-1 r-i where equality for any r =©• kx = ... = k^ . Lemma 2 (Maclaurin): Let k^ 1 =£ i =£ n, HQ and Hr be as in the previous lemma. If Hx, Hs, 1 =S s ^ n are positive then: IX i/ H 2s H 2 3 2s ... 2^ H 1 $ s s£ n 12 3 s where equality at any stage kx = ... = kn . DEF. 18. An umbilic point: Is a point p in a hypersurface Mn where A = XI. where A is the second fundamental form at p, X is a scalar P ' P ' ' and I is the identity transformation on Mp. At an umbilic point all principal curvatures are equal and all directions are directions of principal curvature. If all points of Mn are umbilic, Mn is called totally umbilic. The following important lemma will be used: Lemma J>: A point p e Mn embedded in a riemannian manifold M*"*"1 * of dimension n 2; 2 is umbilic if k, = ... = k at p. x n 64. Proof: The principal curvatures k^ 1 < i ^ n are extremal values of the quantity: A„ e1 e^ / g„ e1 e^ at p for an arbitrary tangent vector e = (e1) of Mn. Hence if a ... = k at p then the given ratio is independent of e so A. . = eg.. V i} j at p, where c is a scalar invariant. Hence p is umbilic. DEF. 19. Riemann n-sphere: Is a closed hypersurface in a -n+1 / \ n riemannian manifold M (n ^ 2) such that every point of K is umbilic. _n+l Theorem 1: Let M be a riemannian (n+l)-manifold (n ^ 2) with constant riemannian curvature such that there exists a global normal coordinate system at a fixed point 0. Then for a closed orientable embedded C hypersurface M jf (Hr_j_ + Hr p) * 1 = 0 r = 1, n and where rp = Y. e where e ,, is the unit normal n+l n+l vector to at a point q and Y is the position vector of q w.r.t. the global coordinate system. Proof: The procedure is exactly as for theorem B where in this case take 5 = Y and e ,.. = e(Y), and use the (n-l)-form: we n+l ' Bn-1 = I'dY> dY' Y> en+r den+l' den+l' 65. Applying the operator d to this and using the facts that d(dY) = 0 and DE r dY'A ...A /dN Y A/dx e n+1.! 'A • • • n+A1 . 1 = n!(-l)r n+H1 e,*l the result follows immediately by integration and application of Stokes1 theorem. Q.E.D. From this theorem several important characterization theorems may be deduced: Theorem 2 (Feeman - Hsiung): Let M" (n ^ 2) be a hypersurface of a _n+i - riemannian manifold M satisfying the conditions of theorem 1. If there exists an integer s, (1 ^ s ^ n), such that H > 0, and either p ^ -H ,/H or p ^ -H ,/H s ' s-1 s , * s-1 £ at all points of then M" is a riemann n-sphere. Proof Case (i) 1 *S s ^ n-1 The inequalities in the theorem are respectively equivalent to: H p + H t =S 0 and H p + H , ^ 0 since H > 0. s-1 s^ s-1 s Now theorem 1 for r = s, together with either of these inequalities => p = -H i/H ; and on substituting this value of p into S JL S theorem 1 for r = s + 1 gives: 66. 2 f ~- (H - H 1H1)*1 = 0 / H s s-1 s+1 v ' •> s . From lemma 1 for r = s the above integrand is non-negative and hence (^.l) holds iff: 2 H -H , H ,1 =0. s s-1 s+1 Hence from lemma 1. k, = ... = k at all points of Mn, and it then 1 n follows from lemma 3 that Mn is a Riemann n-sphere. Case (ii) s = n: Apply theorem 1 with r = n / (H , + H p) * 1 = 0 J n-1 n M11 and since the integrand is of fixed sign then p = -H 1/H . & * n-1 n Now apply theorem 1 with r = n-1 H * 1 = - H - p * 1 n-2 J n-1 if1 ' M* = r (H ?i H ) * i j n-1 n' 2 or /f~-(H H v n-1 1 -HHn n-2o)*' l = 0 J n 67. and a similar argument to case (i) proves the result. Q.E.D. Theorem 3 (Hsiung): Let M" (n ^ 2) be a hypersurface of a riemannian _n+l manifold M satisfying the conditions of theorem 1. Suppose there exists an integer s, (l ^ s n) such that at all points of p is of the same sign, H. > 0 for 1 ^ i s, and H is constant. Then Mn is a Riemann ' s n-sphere. Proof: Case (i) s < n. From lemma 1 for r = 1, s and the assumption that EL > 0 for i = 1, ..., s we obtain: Hn H„ H ,1 J: > _i > ... > -Jll . H_o H.l Hs In particular H H H (h 2) i s ^ s+1 ' where equality =^ = ... =kn« From theorem 1 for r = 1 and the 11 assumption Ex > 0 and p is of constant sign on all of M , it follows that p is negative. Multiply both sides of C+.2) by p and integrate over Mn and apply theorem 1 for case r = 1 and r = s+lj it thus follows from the assumption that Hg is constant: -H f*l = f HlHsp*l ^ f Hs+1p*l =-Hs / *1 s Mn M" Mn Mn 68. whence (H H H )p 1 =0 ( 3) / i s " s+l * ^ Now (4.1) this integrand, is non-positive and hence H. H - H .. = 0 i s s+1 whence by lemma 1 we have: k, = ... = k on all of and in application of lemma 3 proves this part. Case (ii) s = n: Using the assumption H^>0 (l^i^n) and lemma 2 we have: H, S H* S ... S H^"^ & H = c (k.k)v i 2 n-1 n ' where c = constant > 0. From theorem 1 for r = n and inequalities (4.4) we have: n-1 f Hn p * 1 = -| H^M < -c 1 (4.5) Also from theorem 1 for r = 1 and inequalities (4.4) and the fact that p < 0 we have: 11 1 1/n fnn P * l = c " f Hn P * l (4.6) ^ c: n_1 | I^pM = -c""1 f * 1 69. From (^.5) and (h.6) we thus have: l/n I (Hn -1^)5*1 = 0 M*1 Also (^.*0 ==> this integrand is non-negative and hence H l/n = H n 1 whence lemma2 —t> k, = ... = k on all Hence by lemma 1 the in result follows. Q.E.D. _n+l In the case where M is Euclidean (n+l)-dimensional space E and K is a convex Q hypersurface the integral formula in theorem 1 was obtained by Minkowski [22 ] for the case n = 2 in the well-known form J* (Kp + H) * 1 = 0 where H and K are the mean curvature and Gauss curvature respectively at a point q e M2, and p is the oriented distance from a fixed point 0 e E3 to the tangent space Jl2 of the ovaloid M2. The theorem was extended for general n by Kubota [23]. _n+l n+i n Also in the case where M is E and M is a convex C hypersurface theorem 2 with the restriction that s = 1 and the 70. equality on p: H i p + -2=! = 0 Hs being satisfied - (rather than the two corresponding inequalities in theorem 2) - was proved by Grotemeyer [2h] for the case n = 2: 2 ik 2 f (H -K)p*l = ± f g Hk(x ). *1 where p is distance from tangent plane to the origin, x2 is the square of the radius vector and (x2)^, H^. the derivatives of x2 and H w.r.t. the surface parameters. Again this was extended to the case general n by Sfiss. Finally theorem 3 is essentially the well-known Liebmann-Suss theorem, which was obtained by Liebmann [ 1 ] for the case n = 2 in the form: Liebmann H-theorem: The only ovaloids with constant mean curvature H in E3 are the spheres. The extension of this theorem to a convex hypersurface in En was given by Suss [ 3 ]. Finally we note that by using different methods Hsiung [7 ] obtained theorem 1 for the cases r = 1 and r = n and some special cases of theorems 2 and 3j and also theorem 1 for 1 ^ r ^ n, 71. together with theorems 2 and 3 in the special case of a c2 (instead of the c3 condition of theorems 2 and 3) hypersurfaces in Euclidean space. Theorem k (Stong): Under the assumptions of theorem 1, if there are integers s and i (l ^ i < s ^ n), with H , ...,H. > 0 s x and constants c. ^ 0 for i ^ j ^ s-1 such that on all 3 of M11 we have H = Sc.H. then Mn is a Riemann s 3 3 n-sphere. Proof: From theorem 1 H. H. n / H. N / H n H. , 5= 0 H Tr ^ H .. ' ^ H H. _a _ 3-1 = ( i_ ^ f s-i _ 3-1 s H 1 s-1 s j s-1 ° i =S j « s-1 and equality holds everywhere only if M31 is a Riemann n-sphere. Hence: i = ) cjiUVcYil 3 \ H / ^ 3 ^ H , s s-1 or H n - \ c. H. , =5 0 s-1 D J-l and again equality holds everywhere only when is a Riemann n-sphere From theorem 1: 72. /(HB-I-ICJVI)#1 = "/P(H--ICJVM = 0 so that H , = \ c. H. , on all of M" Q.E.D. Theorem 5 (Stong): Under the assumptions of theorem 1, if there exist integers s and i, (0 i < s < n), with H .... H. , > 0 and constants c. 3= 0 for i =Su n ^ s-1 s+1' ' 1+1 j such that at all points of H a Ec.H, and if p is of fixed sign on all of then MN is a RLemann- n-sphere. Proof: Apply procedure of previous theorem in reverse H - ) c. H... ^ 0 s+1 o j + 1 and equality only if Mn is a RLemann n-sphere. By theorem 1 = 0 and since the integrand on the left hand side is of fixed sign H , , = Ec.H.,, everywhere. s+1 3 J+l Q.E.D. 73. Theorem 6: If there is an integer s, (l < s n), with H > 0 and s a constant c with H = c H at all points of M.n , then s s—J. is a Riemann n-sphere. Proof: H > 0 —s c 4 0 and H .. must be fixed in sign. From s ^ s-1 lemma 1 s-1 s-1 s-2' s-1 s s-2 So H , - cH 0 is of fixed sign and vanishes identically only if M*1 is a Riemann n-sphere. Theorem 1 gives: / (Hs-l -cHs-2> *X = / ^cHs-l -Hs^* 1 = 0 and this implies: H , = cH „ on all Mn s-1 s-2 Q.E.D. Theorem J: If there is an integer s (1 < s =£ n) with H. > 0 for (1 ^ i < s) and a constant c with: l/(s-l) 1/s H , ^ c ^ H s-1 s on all of Mn and if p is of fixed sign on then M" is a Riemann n-sphere. 7*. Proof: p of fixed sign and theorem 1 with r = 1 p < 0. By- lemma 2, H 5= c. Choose the orientation of for which g 3= 0 throughout Mn / g * 1 ^ 0. Then theorem 1 gives: - M11 * -/Hap*l = H^*! ^ ^ c8"1 * 1 = c^p^ * 1 all terms are equal and: ^vfa1 - c) * 1 =* 0. So H = c. By theorem 5 with s = 1 and i = 0 we get the required result. Q.E.'D. Theorem 8:(Stong): If there is an integer s (1 < s n) with H , ' • ' ' ' • • 5 H 1 > 0 and a constant c with: s-1 H . H 0 s-1 ^ ^ s-2 > c ^ H H , s s-1 75. on all and if p is of fixed sign throughout M then M11 is a Riemann n-sphere. Proof: Prom theorem 1 with r s s we have that p < 0. Choose the orientation as in the previous proof, then from theorem 1 we get: V2*1 - -\ PVI*1 =2 - I pcHg * 1 = / cHg_1 * 1 Hence all these terms are equal, so /p(Hs-l - CV * 1 - 0 11 and p(H 1 - cH ) =£ 0 H . = cH on all of M . Hence by S"™4. S S"*X s theorem 6, Mn is a Riemann n-sphere. Q.E.D. Corollary: If Z is a closed orientable surface of class c3 twice differentiably embedded in E3 with Hg > 0, then either 76. inf H.2 < Sup or £ is a sphere. qe£ qe£ Proof: Hadamard's theorem £ is the boundary of a convex body, thus, choosing the origin to be inside the body, p is of fixed sign. 2 If inf Hj 5= Sup H2 3 a constant c > 0 such that H,2 5= c2 5 H 1 2 Since H is continuous, either ^ c or H± ^ -c. In the first case theorem 7 implies every point of £ is umbilic =^ £ is a sphere (see Willmore [25] , p. 128). In the second case p must be positive and choosing the orientation as before: -J* cpHx * 1 = J" c * 1 =s - Hx* i = y PH2*1 £ £ p c2 * 1 *£ - / cp Hi * 1 77. Thus all the terms are equal and we have: H2 = -cHl By theorem 6 every point is umbilic and £ is a sphere. Q.E.D. § 2. Characterisation in arbitrary co-dimension The characterisation in the previous section resulted from a specialisation of the integral formula (3«8) to the case of co- dimension one. In this section we shall study the question as to what similar characterisation is possible in arbitrary co-dimension. In what follows we shall assume that the vector e(£) coincides with the Euler-Schouten vector e (i.e. the unit vector in the direction of the mean normal curvature vector); we shall further assume that the vector field I is, at each point of Mn, contained in the (n+l)-dimensional space spanned by the tangent space to Mn and the vector e. In this case the integral formulas (3*5) and (3.8) become respectively: Trace f*(£g) n(p p(i) * i + *1 = 0 (1^7) 2n g L g and H PU)*I + Trace C(^ ) f* (fig) * 1 = 0 r+1 2n M n M n (^.8) 78. and in case the group T generated by | is conformal these become respectively: f H(* ) p (S)* 1 + f **1 = 0 (^.9) and H e / (r+l)p(0*l+ f *E( r)*l = 0 (k.10) By analogy to DEF. 19 we now define: DEF. 20. An Euler n-sphere: Is a closed orientable embedded n- dimensional 'submanifold such that every point is umbilic with respect to the Euler-Schouten vector. _n+N Theorem: Let M be an (n+N)-dimensional (n+N ^3) C riemannian space form*, admitting a C° one-parameter group of conformal transformations generated by a vector field £ ; and Mn be a closed orientable embedded submanifold. If p(|) is positive (or negative) at each point of Mn and M*1 has constant (first) mean curvature PI then M" is an Euler n-sphere. Proof: Multiply (^.9) the constant H, giving: * i.e. riemannian manifold of constant curvature. J" H2 p(|) * 1 + 4> H * 1 a 0 From (^i-.10) with r a 1 we have: f H(|) p(£) * 1 + jT *H * 1 = 0 Hence jV -H(|)) p(i) * 1 =0 Whence from the assumption on p(£) this is true iff H2 -H(*) = 0 since H2 ~H(!) = — V (X - X )2 ^ 0 2 1 n2(n.i) L J Hence at each point of Mn we have: k = k = ... = k 12 n Hence each point of Mn is umbilic with respect to e. 80. _n+N Theorem: Let M be an (n+N)-dimensional (n+N ^3) C riemannian space form, admitting a C° one-parameter group of conformal transformations generated by a vector field t ; and ^ be a closed orientable embedded submanifold. If p(£) is positive (or negative) on M11, and if the principal curvatures k.. , k , ..., k at each point of Mn are positive and H(^) is constant for any r (l < r < n-l) then is an Euler n-sphere. Proof: Multiply (^-9) by the constant H(^ ) and we get: f HH(^) p(£) * 1 + jT •H(^) * 1 - 0 (U.ll) Hence from (^.10) and (^.ll) we get: From the assumption on p(£), this holds iff HH(^) -H^) = 0 since / e \ r' (n-r-l)i e HH( ) - H ( ... J = > k. ... k. (k - k J.1 f r+ r r+1 V V n (nI) L H ir-l 25 0 . 81. Thus at each point of we obtain: k1 = 1^ = ... = kn Whence each point of Mn is umbilic with respect to the direction e. Hence Mn is an Euler n-sphere Q.E.D. Theorem: Let M be an (n+N)-dimensional (n+N ^ j) G riemannian space form admitting C° one-parameter group of conformal transformations generated by a vector field £ , and be a closed orientable embedded n-dimensional submanifold. If on all of the following hold: (i) p(£) is positive (or negative) (ii) Hp(|) + • ^ 0 (or ^0) then Mn is an Euler n-sphere. Proof: We can rewrite (^.9) as: jT (Hp(6) + *) * 1 = 0 Then from the assumptions we have: Substitute this into (^.10) with r = 1. Whence we have: 82. 2 f(E -H(*) ) p(|) * 1 = 0 and the required result follows.. Q.E.D. -n+N oo Theorem: Let M be an (n+N)-dimensional (n+N S= 3) C riemannian space form admitting a C° one-parameter group T of conformal transformations generated by a vector field £ , and Mn be a closed orientable embedded n-dimensional submanifold. If (i) H is positive (or negative) on all of Mn (ii) T is such that on all of Mn. Then M11 is an Euler n-sphere. Proof: Rewriting (U.9) as follows: f H(pU) + | ) * 1 = 0 Hence from the assumptions H > 0 (or < 0) and p(£) + - 3= 0 II (or =S 0) on all M11 we have: 83. P(i) +H = 0 Substituting this in (^.10) with r = 1, we have: 0 2 H -H(|) ) 0 H It now follows from our assumptions that this is true iff: H2 -H(*) = 0 Whence the required result follows. Q.E.D. § 3. Generalisations of § 1 In all our work so far we have been restricted to ambient manifolds of constant riemannian curvature. It is clearly of interest and importance to investigate to what degree this restriction can be removed and/or replaced by weaker assumptions on the ambient space. To this end we have the following theorem due to Feeman- Hsiung. _n+l Theorem $\ Let M be a riemannian manifold of dimension n 3= 2 such that there is a normal coordinate system of Riemann _n+l at a fixed point 0 covering the whole of M . Then for n a closed orientable hypersurface M of class C embedded 8k. n-i-1 in M H , * 1 + H p * 1 = 0 0+.12) r-l L r M M r is an odd. integer less than or equal to n, and p is as defined in theorem 1. Proof: The proof is exactly the same as that of theorem 1 except that this time r is an odd integer and the vanishing of the fourth member of the derivative of the right hand side of the form B , n-1 in theorem 1 is due to the pairwise cancellation of its terms. Q.E.D. From this we have immediately: Theorem (Feeman-Hsiung): Let M11 (n 3= 3) be a hypersurface on a _n+l riemannian manifold M satisfying the conditions of theorem 9» Suppose that there exists an odd integer s, (l < s < n) such that > 0 for i = s, s-1, s-2, and either p ^ -H Jn or p ^ -H JYL at all points S -L S S*-X s of Mn. The Mn is a Riemann n-sphere. Proof: By the same argument as in the proof of theorem 2, equation (^.12) for r = s, together with either of the two inequalities: 85. Hp+H , 0 andHp+H . 3= 0 =>p = -H J E . s s-1 s s-1 s-1 s Substituting this value of p in equation (^.12) for r = s-2, -which is an odd integer by assumption we obtain: H H H H 1 0 TB (, s-5 s - s-2 s-1 ) * - Since H , and H are positive, from lemma 1 for r = s-1 and s-1 s-2 r s= s-2 we obtain: s-3 s-2 ^ s-1 H 0 H - H s-2 s-1 s from which it follows that H H H H s-5 s - s-2 s-1 * ° ^ Thus the integrand on the left hand side of (^.13) is non-positive, and the equality in (^.1^) holds. From lemma 1 it follows immediately that ls.x = ... = k at all points of and thus by lemma 3 "the theorem is proved. Q.E.D. Theorem: Let M" (n ^ 3) a hypersurface on a riemannian manifold _n+l M satisfying the conditions of theorem 9- If there exists an even integer s, (l and p ^ -H / H at all points of V^, then Mn is a Riemann S i. s n-sphere. Proof: By lemma 1 p ^ -H / H =S -H „ / H 1 and by theorem ^\ f ^Hs-2 + PVl)#1 = ° p ^ -H . / H ^ -H J H . = x p so s-1 s s-2 s-1 Thus H _j H _2 — = at all points of M11 H H . s s-1 and by the lemma M*1 is a Riemann n-sphere. Q.E.D. Theorem: with similar conditions to previous theorem. If there > an< is an even integer s} (l < s < n) with Hg, ^s+i 0 * 11 n p > -H 1 / H at all points of M , then M S J. s is a Riemann n-sphere. Proof: By lemma 1 , p 5= -H , / H ^ -H / H .. and by theorem 9: (H + pH / s s+i) * i - o M" 87. so H /H H /H 53 P » - s-1 s » - s s+1 P Therefore H - / H =H/HJ.1 s-1 s s s+1 at all points of Mn, thus Mn is a Riemann n-sphere. Q.E 88. CHAPTER V In this chapter we shall obtain, by the methods already developed, an integral formula for a pair of compact embedded sub- manifolds. We shall then use this integral formula to find conditions under which a volume-preserving diffeomorphism between the submanifolds is an isometry. This generalises somewhat a similar theorem by Chern and Hsiung [17]. § 1. Algebraic Preliminaries Let V be an n-dimensional vector space with a bilinear functional G : V X V -> TR on it. If {e^ ..., e } is any basis, then G is completely determined by the values: %a = G(ei' V 1 J * If we change the basis with a matrix T • (t^) such that e = Te then the matrix G = (g..) is changed according to t t T G T T denotes transpo; If H is a second bilinear functional and h. . = H(e., e.) ij 1 y 89. we can consider the determinant: det(G + XH) =* det G + n\P(g. ., h. .) + ... + Xn det H . Note that under a change of basis the ratio of any two coefficients in this polynomial will be multiplied by (det T)2 and is thus independent of the choice of basis. In particular if G is non- singular: HG := P(gij' hij5/ det G t5*1) depends only on G and H. Clearly in case G = I « id w » ( i ) Trace H = - \ h. . . u n ' n / , 11 Since the construction (5.1) is linear in H, a similar construction can be carried out in the case that H is a vector-valued bilinear function on V. We shall have recourse to the following simplified form of a lemma due to Lars Garding [2 6] . Lemma: Let G and H be symmetric positive-definite bilinear real- valued functions over V x V. Let g « det G and h = det H. Then h 1/n H G g where equality holds iff H = pG for some p. 90. § 2. Geometric Preliminaries n 1 fi Let (Mx , g1, fl-jj, (M^ } g2) 2) be two C°° riemannian 00 n-mamfolds, and f : Mx -> be a C immersion. On M1 there are now two connexions - firstly the riemannian (metric) connexion obtained from and secondly the induced connexion derived from the induced metric f*g2. The second of the two connexions can also be obtained in the usual way, simply by using the pull back of the riemannian (metric) connexion on M2, i.e. the second connexion is simply: f * °^ . The difference between these two connexions on M is also a connexion, which we denote simply by D = (D^). By means of the construction in § 1 we can, using D, construct a vector field D(gL) on M as follows: f n e D(gl) := gf^X) - ( * ^i(\» j (5.2) Let X = (x1) be a tangent vector field on M^. The divergence of X (denoted div X) can be defined in (at least) two entirely equivalent ways; on the one hand, as the function on defined by: X(*l) - (divX)*l (5.3) where *1 is a fixed volume element on 1^. We are of course assuming implicitly that M is orientable. - It follows immediately from the 91 definition of Lie derivative that (5-3) is equivalent to: (divX)*l = £(*l) X on the other hand, on a local coordinate neighbourhood U, with local coordinates (u1, u11), the above definitions take the more classical form: divX = ~ Y —r (Vg X/) (5A) 1 1 1 >Tgi L, du \ / where X = X. d/cki1 on U. In terms of covariant derivatives we l can write (5«*0 simply as: divX = g1D X. . 1 i>J DEF. 21. By means of the induced metric on M, (i.e. f* g ) we J- 2 can define an analogous function which we shall call the induced divergence with respect to mapping f of the field X, and we shall denote it simply by div(f) ij div(f) X := (f*g2) X . Consider, now two embeddings xx, ^ of an abstract n-dimensional _n+W orientable differentiable manifold into a riemannian manifold M 92. of dimension n + N. Further let f : x1(M) -» ^(M) be a diffeo- morphism. We thus have the commutative diagram as follows: xx(M) <= M M I f x2(M) ez M We call the mapping f an isometry if at a point p e M < dxjp), dx1(p) > . < dxjp), cbc^p) > (5.5) i.e. if it maps the induced riemannian metric of the embedded sub- manifold (M, x1) into the induced riemannian metric of the embedded submanifold (M, x ). In terms of the cotangent map f* we can write (5.5) simply as: Over the abstract manifold M we now have two metrics namely gx : < dxx(p), dxJ_(p) > and g2 : < dx2(p), dx2(p) > = < d(foXl)(p), d(f oXjL)(p) > In what follows, unless it is obvious or specifically stated to the contrary, we will work on the manifold M-,_ with the (induced) metric 95- The notion of frames elf having measure 1 and coherently orientated with that of M thus has a sense in both metrics. At a point p e M any such frame can be obtained from a fixed one by a linear transformation of determinant 1. The induced map dx^ on the tangent spaces is univalent, so we can identify e^ and 1 1 dxx(e^). Let to , a/ be the dual coframe of ex, e , then the volume element on M is: * 1 = enA ... A CO We call the above mapping f volume-pre serving if it maps the volume element of one manifold into that of the other, i.e, -p* f where cjdx and cd2 are respectively the volume elements of Mx and M^. It follows immediately from the definition that a volume- preserving diffeomorphism exists only if M is oriented and the diffeomorphism f is then oriention-preserving. § 5» Integral Formula Let (M,g) be a C°° riemannian manifold of dimension (n+N), oo r\ oo which admits a C vector field |. Further let W be a C n- 00 dimensional orientable manifold with two compact C immersions x± and into M; and let f : xx(M) -» x2(M) be a volume-preserving diffeomorphism. Assume £ lies nowhere entirely in the tangent space of xi(M). We shall work with the bundle ATM over xx(M). We shall write simply Mx, M2 respectively for x^M), ^(M) . As seen in § 1 the manifold M1 inherits a metric g1 from M as does TA2 inherit a metric g2. On M1 we thus have the two metrics: At points of M we can thus represent the vector field £ Hence k Pjl) = S gM = t± (5.6) where PJD ••= < I) e. > . Thus < dl, e. >_ + < I, de. >- • d£. ' 1 g ' 1 g 1 < d.S, e. >- + < i, d.e. >- m d.6. (5.7) 95. Now from (1-13) we have: -j k , a de. = a). e, + co. e l i k i a d. e. =s cD.^fe.ie, + co.a(e.)e J i i J k i j' a whence from (5»7) a < d.g, e. >_ + < |, ai. (e.)e. >_ + <£, CD. (e.)efT >- J ig ' i ^ j' k g ' l x D a g - . (5.8) Now use g -<|, <(e.)ea>_ - g^dX J g Hence The right hand side of this equation is simply div £ , and ^/(e.) = ^^V.) • (from (1.15)) CT Further, the mean curvature H in the direction ea is by definition: Trace A° 96. i.e. HCT := g ^ A°. . = Trace A° 1 1 ^ 6, Thus we have: u Si < ei >g + H pa(|) - div 6 , : e where Pa(^) = < ^ a >g Note: We use H to denote the (first) mean curvature with respect 1 to the (induced) metric g± . Finally the above equation becomes: (!) Trace Xl*( £g) + E° p(S) = div 5 (5.9) Si . i 1 Whence: ^ Trace Xl*(£g) + H° P In exactly the same manner, beginning with (5.8) and by using this time the second induced metric on Mx viz. f*g2 we get ld k (f *g2) < dji, e. >. + (f *g2)^ < |, c. (e.)ek > - 97. If we agree to write simply g1J for (f *g )1J we have 2 2. (i) Trace x *(£g) + g iJ a,k (e )p (g) + Ha p (|) = g ^ d I. J J g2 - £ . - 2 . ff where H is the (induced) mean curvature in the direction ea with 2 1 CT respect to the second induced metric g2 ^, i.e. H° = Trace A . 2 g2 Finally we get: (|) Trace x *( £ g) + HCT p ( g) = div(f)g (5.10) S2 •• £ . 2 °. whence: ^{(i) Trace x/(£g) + E° P (£)} * 1 = f div(f)g * 1 . This extends the corresponding integral formula of Chern and Hsiung [17]. If we now subtract (5.9) from (5.10) and use Stokes' theorem we arrive at our basic integral formula. Theorem 1: If (M,g) is a riemannian manifold admitting a C°° vector field £, and M1, M2 are a pair of compact C°° immersed equidimensional submanifolds which do not contain any singular points of £ , and such that £ is almost nowhere 98. (i.e. at most on a set of measure zero) tangent to M , then for a volume-preserving diffeomorphism f : M -» 1^ we have: (i) / (Trace - Trace ) x *(£ g) * 1 + / (H° - H°)p (4) *1 -J 8 g --6 J 2 1 a- M 2 1 M, = ^ div(f)4 * 1 We note immediately that in case f is an isometry the right hand side of this equation is zero. By analogy to the work of Chern and Hsiung [17], we make the following: DEF. 22. A mapping f for which div(f)£ = 0 is called an almost isometry. § k. Specialisations We now consider the integral formula in case f and £ are specialised. In particular in all that follows we shall assume that f is an almost isometry. In case I is a conformal field: / (Trace gx - 1 ) • * 1 + / (H° - H°) p (I) 0 J 6 J .2 1 a. (5.11) 99- Hence / Trace g (5.12) In case £ is proper nomothetic we have: f Trace g]_ * 1 + -i f (U° - E°) po(6) * 1 = v(M1) (5.13) In case | is isometric: I (H^-H^) po(|) * 1 = 0 (5.1*0 We can now generalise a theorem of Chern and Hsiung ['7] . Theorem: Let xx, x2 : M -» M be two imtaddcd compact submanifolds of riemannian manifold M which admits a conformal vector field £ with the properties of theorem 1. If f has the properties: a (i) (ii) HP ^ K°, 2 1 then f is an isometry. 100. Proof: From the conditions of the theorem and Garding's lemma we get Trace gz - Trace g1 > 0 §2 gi and conditions (i) and (ii) give us 3I 1 = 1 g2 whence Garding's lemma gives us immediately that: g = g i 2 Q.E.D. Note that in case the ambient space is En+^ and £ is the homothetic Killing vector field on En+^ with components ^ = x^, x^ being rectangular coordinates with a point in the interior of Mn as origin, then the orbits of the transformations generated by £ are simply the lines through the origin and we have £g = 2g and our integral formula of theorem 1 reduces to that of Chern and Hsiung [17]. 101. CONCLUSION In the foregoing we have not concerned ourselves with the global existence of linearly independent sections of the normal bundle. Although in a neighbourhood of a point there are linearly independent sections, the existence of a single section globally is an open question. Work in this direction can be found in Stiefel [28], Hirsch [29] (who showed that the immersion problem is essentially a cross-section problem for the normal bundle), Kervaire [30] and Handel [31]. Problem 1: It is clear from the foregoing work that the mean normal curvature vector contains much important geometric information about the immersion/embedding. It is clearly desirable in the future to study its properties in more detail. An obvious generalisation of T) is: the r**1 mean normal curvature vector Tj(r), defined: Tj(r) := H(^)eQ (C.l) 1 where H( ° ) is the r^* mean curvature in the normal direction ea to the submanifold. Clearly rj1) is just the classical mean normal curvature vector. We can thus ask: what are the analogous properties of T}(r)? Problem 2: Do analogues of theorems A and B exist for T](r)? Problem g: In Chapter IV we have been restricted to a space of 102. constant curvature, we can thus ask: to what extent can this condition be relaxed and/or replaced? Theorem 9 shows that in certain cases at least it is redundant. Problem hi What is the nature of the umbilic set of a submanifold? Can_it only be either: (i) a discrete set or (ii) the submanifold/nul set ? Problem 5; What is the connection between curvature and the umbilic set? Concerning this we have the following example: consider a 3-dimensional space M3 with fundamental form: = y 2 =c c nstant ds2 (X ^ ^ ) X ° i=l 0=1 ( ^ 2) Calculation shows that M3 - (0, 0, 0) is not of constant curvature and that every point of the surface in M3 given by: yx = sin u cos v y = sin u sin v is umbilic. J 2. y3 = cos u Problem 6: In Chapter V we have been concerned with a volume-preserving diffeomorphism. It is thus important to ask: what restrictions does the existence of such a diffeomorphism impose on the ambient manifold? 103. 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