Morse Theory by J

MORSE THEORY BY J. Milnor Based on lecture notes by M. SPIVAK and R. WELLS

PRINCETON, NEW JERSEY

PRINCETON UNIVERSITY PRESS

1963 CONTENTS Copyright 0 1963. by Princeton University Press All Rights Reserved PART I . NON-DEGENERATE SMOOTH FUNCTIONS ON A MANIFOLD L . C . Card 63-13729 $1 . Introduction...... 1 $2 . Definitions and Lemmas ...... 4

$3 . Homotopy Type in Terms of Critical Values ...... 12

$4 . Examples ...... 25 $5 . The Morse Inequalities...... 28 06 . Manifolds in Euclidean Space: The Existence of

Non-degenerate Functions ...... 12 $7 . The kfschetz Theorem on Hyperplane Sections ...... 39

PART I1 . A WID COURSE IN RIEZ"NIAN GEOKETRY $8. Covariant Differentiation ...... 43

$9. The Curvature Tensor ...... 51 $10 . Geodesics and Completeness ...... 55

PART I11 . THE CALCULUS OF VARIATIONS APPLIED TO GEODESICS $11 . The Path Space of a Smooth Manifold ...... 67

§12 . The Energy of a Path...... 70 $13 . The Hessian of the Energy Function at a Critical Path . . 74 $14 . Jacobi Fields: The Null-space of E*++...... 77 515 . The Index Theorem ...... a3 §16 . A Finite Dimensional Approximation to QC ...... aa 817 . The Topology of the Pull Path Space ...... 93 $18 . EXistence of Non-conjugate Points Printed in the United States of America ...... 9a §I9 . Some Relations Between Topology and Curvature ...... loo

V CONTENTS

PART IV. APPLICATIONS TO LIE GROUPS I IC SPACES ...... $20. Symmetric Spaces 109 ...... $21. Lie Groups as Symmetric Spaces 112

$22. Whole Manifolds of Minimal Geodesics ...... 118 $23. The Bott Periodicity Theorem for the Unitary Group ... 124 $24. The Perjodicity Theorem for the Orthogonal Group .... 13’1

PART I ...... 149 APPmIX. THE HGMOTOPY TYPE OF A MONOTONE UNION NON-DEGENERATE SMOOTH FUNCTIONS ON A MANIFOLD.

$1. Introduction.

In this section we will illustrate by a specific example the situ-

ation that we will investigate later for arbitrary manifolds. Let us consider a torus M, tangent to the plane V, as indicated in Diagram 1.

Diagram 1.

Let f: M+R (R always denotes the real numbers) be the height above the V plane, and let I@‘ be the set of all points x E M such that f(x) 5 a. Then the following things are true:

(1) If a < 0 < f(p), then M& is vacuous.

(2) If f(P) < a < f(q), then I@’ is homeomorphic to a 2-cell. (3) If f(q) < a < f(r), then @ is homeomorphic to a cylinder:

vi (4) If f(r) < a < f(s),w then M” is homeomorphic to a compact manifold of genus one having 6. circle as boundary: §I. INTRODUCTION 3 2 I. NON-DEGENERATE FUNCTIONS

The boundary

i.k = (x E Rk : ((x((= 11

willbe denoted by Sk-' . If g: Sk-' -+ Y is a continuous map then (5) If f(s) < a, then M& is the full torus.

In order to describe the change in Mi" as a passes through one Y vg ek convenient consider homotopy of the points f(p),f(q),f(r),f(s) it is to (y with a k-cell attached by g) is obtained by first taking the topologi- In terms of homotopy types: type rather than homeomorphism type. as far as ca1 sum (= disjoint union) of Y and ek, and then identifying each is For (1) + (2) the operation of attaching a 0-cell. x E Sk-' with g(x) E Y. To take care of the case k = o let eo be a homotopy type is concerned, the space Ma, f(p) < a < f(q), cannot be dis- point and let 6' = S-l be vacuous, so that Y with a 0-cell attached is tinguished from a 0-cell: just the union of Y and a disjoint point. - As one might expect, the points p,q,r and s at which the homo- G3 topy type of M& changes, have a simple characterization in terms of f. They are the critical points of the function. If we choose any coordinate Here "J" means "is of the same homotopy type as." system (x,y) near these points, then the derivatives and ar are

(2) + (3) the operation of attaching a 1-cell: is both zero. At p we can choose (x,y) so that f = x 22+ y , at s so 22 that f = constant -x - y , and at q and r so that f = constant + n 22 x - Y . Note that the number of minus signs In the expression for f at

each point is the dimension of the cell we must attach to go from Ma to Mb, where a < f(point) < b. Our first theorems will generalize these W facts for any differentiable function on a manifold.

is again the operation of attaching a 1-cell: (3) -+ (4) REFEFENCES For further information on Morse Theory, the following sources are extremely useful. M. Morse, "The calculus of variations in the large,'' American Mathematical Society, New York, 1934.

H. Seifert and W. Threlfall, "Varlationsrechnung im Grossen," published in the United States by Chelsea, New York, 1951. the operation of attaching a 2-cell. (4) + (5) is 0 R. Bott, The stable homotopy of the classical ~J-OUPS, Annals of The precise definition of "attaching a k-cell" can be given as Mathematics, Vol. 70 (19591, pp. 313-337. follows. Let Y be any topological space, and let R. Bott, Morse Theory and its application to homotopy theory,

ek = (x c~~ : \(XI\< 1) Lecture notes by A. van de Ven (mimeogpaphed), University of Bonn, 1960. be the k-cell consisting of all vectors in Euclidean k-space with length 5 1 I. NON-DEGENERATE FUNCTIONS §2. DEFINITIONS AND LEMMAS 5

since f has p as a critical point.

Therefore f,, is symmetric. It is now clearly well-defined since

$ (w"(f)) = v(w"(f)) is independent of the extension v" of v, while $2. Definitions and Lemmas. P Gn(?(f)) is independent of w". The words "smooth" and "differentiable" will be used interchange- I If (x' , . . . ,xn) is a local coordinate system and v = C ai -1 a ably to mean differentiable of class Cm. The tangent space of a smooth ax: P' w = c bj we can take ij = c b. -a where bj now denotes a con- manifold M at a point p will be denoted by T%. If M+ N is a -$Ip g: J axj stant function. Then smooth map with g(p) = q, then the induced linear map of tangcnt spaces will be denoted by &: TMp -+ TNq. Now let f be a smooth real valued function on a manifold M. A so the matrix ( i-a2f (p)) represents the bilinear function f,, with point p E M is called a critical point of f if the induced map ax axJ If we choose a local coordinate system f,: T% +TRAP) is zero. respect to the basis a (x1 ,...,xn) in a neighborhood U of p this means that We can now talk about the index and the nullity of the bilinear functional f,, on TMp. The of a bilinear functional H, on a vec- The real number f(p) is called a critical value of f. tor space V, is defined to be the maximal dimension of a subspace of V We denote by M" the set of all points x E M such that f(x) a. on which H is negative definite; the nullity is the dimension of the s- If a is not a critical value of f then it follows from the implicit space, i.e., the subspace consisting of all v E V such that H(v,w) = o

function theorem that I@ is a smooth manifold-with-boundary. The boundary for every w E V. The point p is obviously a non-degenerate critical f-'(a) is a smocth submanifold of M. Point of f if and only if f,, on T% has nullity equal to 0. The is called non-degenerate if and only if the A critical point p index of f,, on T% will be referred to simply as the index of f p. matrix The Lemma of Morse shows that the behaviour of f at p can be completely

described by this index. Before stating this lemma we first prove the following: is non-singular. It can be checked directly that non-degeneracy does not depend on the coordinate system. This will follow also from the following LEMMA 2.1. Let f be a Cm function in a convex neigh- intrinsic definition. borhood V of 0 in Rn, with f(0) = 0. Then If p is a critical point of f we define a symmetric bilinear functional f,, on TMp, called the Hessian of f at p. If v,W E T Mp f(Xl,. . .,xn) = i=l then v and w have extensions v" and $ to vector fields. We let* for some suitable Cm functions gi defined in V, with

f,,(v,w) = v" P (w"(f)), whe1.e v" P is, of course, just v. We must show that this is symmetric and well-defined. It is symmetric because PROOF : v"P (ij(f)) - $p(v(f)) = [v",?lp(f) = 0

where [?,%I is the Poisson bracket of v" and %, and where [?,$lp(f) = 0

~ * Here G(f) denotes the directional derivative of f in the direction 3. §2. DPINITIONS AND LEMMAS 6 I. NON-DEGENERATE FUNCTIONS 7

Therefore, applying 2.1 to the gj we have mm2.2 (Lemma of Morse). Let p be a non-degenerate critical point for f. Then there is a local coordinate 1 system (7 ,.. .,yn) in a neighborhood U of p with gj (xl ,* a. ,xn) = i i=l” y (p) = 0 for all i and such that the identity 12 f = f(p) - (y ) - ... - (yX)* + (yX+l)* + ... + (y92 for certain smooth functions hij. It follows that holds throughout U, where X is the index of f at p. f(xl,..., xn) = t xixjhij (x, ,. . . ,xn) . i,j=1 PROOF: We first show that if there is any such expression for f, We can assume that since we can write = l(h hij = hji, Eiij then x must be the index of f at p. For any coordinate system 2 ij+ hji)J and then have Kij = Eji and f =ZXXFIij . Moreover the matrix (li (0)) (z’, ..., zn), if ij f(q) = f(p) - (z’(q))2- ... - (zh(q))2 + (zX+’(q))2+ ... + (z”(q)I2 is equal to ( 1 a2f (o)), and hence is non-singular. 2ZF-S then we have There is a non-singular transformation of the coordinate functions (-2 if i=jlh, which gives us the desired expression for f, in a perhaps smaller neigh- 2 if i = j > h , aZi azJ borhood of 0. To see this we just imitate the usual diagonalization proof 0 otherwise , for quadratic forms. (See for example, BirkhoffandMacLane, “A survey of which shows that the matrix representing f,, with respect to the basis modern algebra,“ p. 271 .) The key step can be described as follows. Suppose by induction that there exist coordinates ul, ...,un in a neighborhood U, of 0 so that

f = + (u1)2 5 (UrJ2 + - ... 1 UiUjHij (5, . . * 9%) 2*. i,jIr \ .II \ ‘1. throughout U,; where the matrices (%j(ul,...,un)) are symmetric. After a linear change in the last n-r+l Therefore there is a subspace of T% of dimension X where f,, is nega- coordinates we may assume that Krr(o) f 0. Let g(ul, ...,%) denote the square root of (qr(ul,..., un) 1. This will tive definite, and a subspace V of dimension n-A where f,, is positive be a smooth, non-zero function of ul,. . . ,un throughout some smaller neigh- definite. If there were a subspace of T% of dimension greater than X borhood U, C U, of 0. Now introduce new coordinates v1, vn by on which f,, were negative definite then this subspace would intersect V, ..., which is clearly impossible. Therefore X is the index of f,,. vi = ui for i f r 1 We now show that a suitable coordinate system (y ,...,y n, exists.

Obviously we can assume that p is the origin of Rn and that f(p) = f(0) =

By 2.1 we can write -n It follows from the inverse function theorem that vl, ...,vn will serve as coordinate functions within some sufficiently small neighborhood U3 of 0. It is easily verified that f can be expressed as

for (xl,. . . ,xn) in some neighborhood of 0. Since 0 is assumed to be a critical point: af gj(o) = -p) = 0 . ax P2. DEFINITIONS m LEMMAS 9 8 I. NON-DEGENERATE FUNCTIONS thoughout u3' his completes the induction; and proves Lemma 2.2.

COROlL&3y 2.3 Non-degenerate critical points are isolated.

Examples of degenerate critical points (for f'unctions on R and

R2) are given below, together with pictures of their graphs.

(d) f(x,y) = x2. The set of critical points, all of which are degenerate, is the x axis, which is a sub-manifold of R2.

7 (b) F(x) = e-'ix2sin2(1/X) (a) f(x) = x3. The origin . is a degenerate critical point. The origin is a degenerate, and non-isolated, critical point.

(e) f(x,y) = x2y2. The set of critical points, all of which are degenerate, consists of the Union of the x and g axis, which is not even a sub-manifold of R2. We conclude this section with a discussion of 1-parameter groups of diffecpnorphisms. The reader is referred to K. Nomizu,"Lie Groups and Differ- ential Oeometrg," for more details. A 1-parameter group of diffeomorphisms of a manifold M is a C"

f(x,y) = x3 - 3xy2 = Real part of (x + i~)~.

(0.0) is a degenerate critical point (a "monkey saddle"). 10 I. NON-DEGENERATE FUNCTIONS 92. DEFINITIONS AND LEMMAS 11

such that Thus for each point of M there exists a neighborhood U and a number E > 0 so that the differential equation 1) for each t E R the map cpt: M -+ M defined by

cpt(q) = cp(t,q) is a diffeomorphism of M onto itself, dTt(9) -=x rpo(q) = q 2) for all t,s E R we have cpt+s = cpt 0 cp, cpt(9)' has a unique smooth solution Given a 1-parameter poup cp of diffeomorphisms of M we define for q 6 U, It1 < E. a vector field X on M as follows. For every smooth real valued function The compact set K can be covered by a finite number of such neighborhoods Let E~ > 0 denote the smallest of the corresponding f let U. numbers E. Setting cpt(q) = q for q 6 K, it follows that this differen-

tial equation has a unique solution qt(q) for It( < E~ and for all

This vector field X is said to generate the group cp, q E M. This solution is smooth as a function of both variables. Further- more, it is clear that providing that cpt+s = cpt 0 cp, lt(,1s),lt+s1< cO. LEMMA 2.4. A smooth vector field on M which vanishes Therefore each such mt is a diffeomorphism. outside of a compact set K C M generates a unique 1- parameter group of diffeomorphisms of M. It only remains to define cpt for It1 2 E~.Any number t can

be expressed as a multiple of ~~/2plus a remainder r with PROOF: Given any smooth curve Irl < c0/2 . If t = k(E0/2) + r with k 2 0, set t - c(t) E M

O *.* ' 'E0/2 ' 'r it is convenient to define the velocity vector 't = 'E0/2 'E0/2 dc where the transformation is iterated k times. If k < 0 it is aT TMc(t) 'pE0/2 only necessary to replace cp iterated -k times. Thus cpt dc lim f(t+h) - f(t) Eo/2 by 'p-E0/2 by the identity =(f) = h-r -h. . (Compare $8.) Now let cp is defined for all values of t. It is not difficult to verify that qt is be a 1-parameter group of diffeomorphisms, generated by the vector field X. well defined, smooth, and satisfies the condition cpt+s = cpt Then for each fixed q the curve cp, , This completes the proof of Lemma 2.4

t + cpt(4) RFMARK: The hypothesis that X vanishes outside of a compact set satisfies the differential equation cannot be omitted. For example let M be the open unit interval dcpt(q) (0,l) C R, -a€---= Xcpt(q) ' and let X be the standard vector field d on M. Then X does not generate any 1-parameter group diffeomorphisms of with initial condition cpo(q) = q. This is true since of M.

where p = cpt(q). But it is well known that such a differential equation, locally, has a unique solution which depends smoothly on the initial condition. (Compare Graves, "The Theory of Functions of Real Variablesi'p. 166. Note that, in terms of local coordinates u' , . . . ,un, the differential equation takes on the more familiar form: & = xi(,' ,...,un), i = 1 ,..., n.) I. NON-DEGENERATE FUNCTIONS 12 $3. HOMOTQPY TYPE 13

C: R + M is a curve with velocity vector $ note the identity =Tdc d(foc) .

$3. Homotopy Type in Terms of Critical Values. Let p: M+R be a smooth function which is equal to

Throughout this section, if f is a real valued function on a 1 / throughout thecompact set f-l [a,b]; and which vanishes manifold M, we let outside of a compact neighborhood of this set. Then the vector field X, M& = f-’(- m,al = (p E M : f(p) 5 a) . defined by

THEOREM 3.1 . Let f be a smooth real valued function Xq = p(q) (grad f)s on a manifold M. Let a < b and suppose that the set satisfies the conditions of Lema 2.4. Hence X generates a 1-parameter f-l [a,bl, consisting of all p E M with a L f(p) L b, group of diffeomorphisms is compact, and contains no critical points of f. Then

M& is diffeomorphic to Mb. Furthermore, is a de- vt: M + M. formation retract of Mb, so that the inclusion map For fixed q E M consider the function t+ f(cpt(q)). If Tt(q) Ma -+ Mb is a homotopy equivalence. lies in the set f-’[a,bl, then down to M& along the orthogo- The idea of the proof is to push Mb .dtdf(Vt(q)) = <-, dcpt(q) grad f> = = + 1. nal trajectories of the hypersurfaces f = constant. (Compare Diagram 2.) Thus the correspondence

t -* f(vt(9))

is linear with derivative +1 as long as f(cpt(q)) lies between a and b. \ Now consider the diffeomorphism M+ M. Clearly this carries M& diffeomorphically onto Mb. This proves the first half of 7.1. Define a 1-parameter family of maps \ I I I I I IIY rt: M~ 4 ~b

Diagram 2. 14 I. NON-DEGENERATE FUNCTIONS § 3. HOMOTOPY TYPE 15

Choosing a suitable cell ex C H, a direct argument (i.e., push-

in@; in along the horizontal lines) will show that MC-E~ ex is a deformation retract of M~-~u H. Finally, by applying 3.1 to the function F and the region F-1 [C-E,C+EI we will see that M~-~u H is a deformation retract

of MC+& . This will complete the proof.

Choose a coordinate system u 1 ,...,un in a neighborhood U of p 90 that the identity

f = c - (U1l2- ... - (uA)2 + (UX+l)2, ... + (Un)2

Diagram 3. holds throughout U. Thus the critical point p will have coordinates

u 1 (p) = ... = u”(p) = 0 . THEOREM 3.2. Let f: M+ R be a smooth function, and let p be a non-degenerate critical point with index A. Set- Choose E > 0 sufficiently small so that ting f(p) = c, suppose that f-’[c-E,c+E~ is compact, (1 ) The region f-’ [c-E ,C+E 1 is compact and contains no critical and contains no critical point of f other then p, for some E > 0. Then, for all sufficiently small E, the set points other than p. MC+E MCFE with a k-cell attached. has the homotopy type of (2) The image of U under the diffeomorphic imbedding

(u1 , . . . ,un) : U -Rn The idea of the proof of this theorem is indicated in Diagram 4, contains the closed ball. for the special case of the height function on a torus. The region “u 1 ,..., un): c (ui)2 5 2El . MC-E = f-l (-m,C-E 1 Now define ek to be the set of points in U with

is heavily shaded. We will introduce a new function F: M -+ R which (u’)~+... + (u‘)~5 E and u’+~= ... = un = 0. coincides with the height function f except that F < f in a small neigh- MC -E The resulting situation is illustrated schematically in Diagram 5. borhood of p. Thus the region F-’ (-m,c-~l will consist of together with a region H near p. In Diagram 4, H is the horizontally shaded region.

fr c’ f C+E I f.CiE ‘f.c

Diagram 4. Diagram 5. 13. HOMOTOPY TYPE 16 I. NON-DEGENERATE FUNCTIONS

F coincide. Within this ellipsoid we have The coordinate lines represent the planes ux+l= . . . = un = 0 and 1 F 5 f = c-c+q -< c+ ls+q2-< C+E u = ... = ux = 0 respectively; the circle represents the boundary of the f-' (c-E) ball of radius G;and the hyperbolas represent the hypersurfaces This completes the proof. and f-' (c+E) . The region MC-€ is heavily shaded; the region f-l [c-E,c] ASSERTION 2. The critical points of F are the same as those of f. is heavily dotted; and the region f-'[c,c+el is lightly dotted. The hori- x PROOF: Note that zontal dark line through p represents the cell e . aF = -1 - Llf(E+2q) < 0 Note that eh n MC-E is precisely the boundary &', so that ex

is attached to MC-E as required. We must prove that MCeE u ex is a de-

formation retract of M'+~. Since Construct a new smooth function F: M - R as follows. Let p:R-R where the covectors dk and dq are simultaneously zero only at the origin, be a C" function satisfying the conditions it follows that F has no critical points in U other than the origin.

E U(0) > Now consider the region F- 1 [c-E,c+E]. By Assertion 1 together u(r) = o for r 2 2~ with the inequality F 6 f we see that -1 < pI(r) o for all r, F-'[C-E,C+EI c f-l [C-E,C+EI . where ~'(r)= Now let F coincide with f outside of the coordinate . Therefore this region is compact. It can contain no critical points of F neighborhood U, and let except possibly p. But

F = f - V((U')~+ (u')~ + Z?(U'+')~+...+~(U~)~) ...+ F(p) = C - P(0) < C - E .

within this coordinate neighborhood. It is easily verified that F is a Hence F-'[c-E,C+EI contains no critical points. Together with 3.1 this well defined smooth function throughout M. Proves the following. It is convenient to define two functions ASSERTION 3. The region F-l (-w,c-E] is a deformation retract of 5,q: u- [o,m) MC+E

by It will be convenient to denote this region F-' (-m,c-E 1 by = (u')2 + ... + (u52 pEu H; where H denotes the closure of F-'(-m,C-El - .

q = (uX+1)2 + ... + (Unl2

REMARK: In the terminology of Smale, the region MC-E u H is Then f = c - 5 + q; so that: described as MC-E with a "handle" attached. It follows from Theorem 3.1 F(q) = c S(q) + q(q) - P(C(q) + 211(9)) - that the manifold-with-boundary MC-E u H is diffeomorphic to MC+E . This for all q E U. fact is important in Smale's theory of differentiable manifolds. (Compare 3. hle, Generalized PoincarB's conjecture in dimensions greater than four, ASSERTION 1. The region F-' (-m,c+~I coincides with the region hlsof Mathematics, Vol. 74 (1961), pp. 391-406.) MCfE = f-l(- ~,c+E1.

PROOF: Outside of the ellipsoid 5 + 2q 1. 2a the functions f and 18 I. NON-DEGENERATE FUNCTIONS 03. HOMOTOPY TYPE

Now consider the cell eh consisting of all points q with

t(q) 5 E, q(9) = 0.

Note that ex is contained in the "handle" H. In fact, since we have

F(q) 5 F(P) < C-E

h but f(q) 2 c-e for q E e .

Diagram 7.

Thus rl is the identity and ro maps the entire region into ex. The fact that each rt maps F-'(-m,c-~l into itself, follows from the inequality aF > 0. 'I CASE 2. Within the region E I. q + E let rt correspond to the transformation

A h+l n (u1 )...,u") -+ (u1 ,...)u ,stu ,...)StU )

where the number st E [0,11 is defined by

St = t + (l-t)((I.-E)/rl)''2 .

Thus rl is again the identity, and ro maps the entire region into the

hypersurface f-l( C-E) . The reader should verify that the functions stui

remain continuous as I. --c E, q -+ 0. Note that this definition coincides Diagram 6. with that of Case 1 when I. = E. The region The present situation is illustrated in Diagram 6. CASE 2. Within the region 11 + E 5 (i.e., within MC-€) let MC-" is heavily shaded; the handle H is shaded with vertical arrows; rt be the identity. This coincides with the preceeding definition when

and the region F-l [C-E,C+E] is dotted. E=q+E.

ASSERTION 4. MC-€ ex is a deformation retract of MC-€ u H. This completes the proof that MC-€u ex is a deformation retract H is Of F-l (-m,C+EI. Together with Assertion 3, it completes the proof of PROOF: A deformation retraction rt: MC-€ u H -. M~-" Theorem 3.2. More precisely indicated schematically by the vertical arrows in Diagram 6. REmK 3.3. More generally suppose that there are k non-degenerate let rt be the identity outside of U; and define rt within U as follows. It in necessary to distinguish three cases as indicated in Diagram 7. critical points p,, ...,pk with indices xl, ...,xk in f-'(c). Then a similar proof shows that MC+" has the homotopy type of MC-E~ e x1 u.. .U e 'k . CASE 1. Within the region 5 5 E let rt correspond to the transformation

(U') ...,un, -+ (u',. . .,uA,tuh+l,... ,tun, . 93. HOMOTOPY TYPE 21 20 I. NON-DEGENERATE FUNCTIONS PROOF: Define k by the formulas REMARK 2.4. A simple modification of the proof of 3.2 shows that k(x) = x for E X the set MC is also a deformation retract of MCCE. In fact MC is a x 1 -A deformation retract of F-' (-m,cI, which is a deformation retract of MC+&. k(tu) = 2t~ for O

is defined by similar formulas. It is now not difficult to verify that the compositions kP and Pk are homotopic to the respective identity maps. Thus k is a homotopy equivalence. For further details the reader is referred to,Lema 5 of J. H. C. Whitehead, On Simply Connected 4-Dimensional Polyhedra, Commentarii Math. Helvetici, Vol. 22 (1949), pp. 48-92.

LEMMA 3.7. Let q: 6'- X be an attaching map. Any homotopy equivalence f: X -Y extends to a homotopy

I equivalence F : x u eA+Y uf'P e A . 'P Diagram 8: MC is heavily shaded, and F-'[c,c+E] is dotted.

PROOF: (Following an unpublished paper by P.Hilton.) Define F THEOREM 3.5. If f is a differentiable function on a manifold by the conditions M with no degenerate critical points, and if each Ma is compact, then M has the homotopy type of a CW-complex, with FIX = f one cell of dimension A for each critical point of index A. i Fie' = identity . kt g: Y --cX be a homotopy inverse to f and define (For the definition of CW-complex see J. H. C. Whitehead, Combin- G: Y ex -x u eA atorial Homotopy I, Bulletin of the American Mathematical Society, Vol. 55, f'P gf'P = = (1949), Pp. 21 3-245.) bg the corresponding conditions GIY g, G(eA identity. The proof will be based on two lemas concerning a topological Since gfcp is homotopic to 9, it follows from 3.6 that there is space X with a cell attached. a homotopy equivalence

k: X u ex --cXu ex . gf'P m LEMMA 3.6. (Whitehead) Let 'po and 'p, be nnotopic maps we will first prove that the composition from the sphere 6' to X. Then the identity map of X ex- GF: x ex. -X eA tends to a homotopy equivalence 'P 'P A k:Xue'+Xue . is homotopic to the identity map. 'PO 'P1 22 I. NON-DEGENERATE FUNCTIONS 03. HOMOTOPY TYPE 23

Step 2. Since G(Fk) 2 identity, and G is known to have a left Let ht be a homotopy between gf and the identity. Using the inverse, it follows that (Fk)G 2 identity. specific definitions of k, G, and F, note that Step 3. Since F(kG) 2 identity, and F has kG as left inverse kGF(x) = gf(X) for x E X,

1 ax also, it follows that F is a homotopy equivalence. This completes the kGF(tu) = 2tU for'o 5 t L~,u E e , 1 .h proof of 3.7. kGF(tu) = h2-2tcp(u) for t 1, u E e . PROOF OF THEOREM 3.5. Let c, < c2 < c3 < ... be the critical The required homotopy values of f: M -R. The sequence (cil has no cluster point since each qT: x ex - x ex cp cp M& is compact. The set M& is vacuous for a < c,. Suppose is now defined by the formula a 4 c,,c2,c3,... and that Ma is of the homotopy ,type of a CW-complex. kt c be the smallest ci > a. By Theorems 3.1, 3.2, and 3.3, MCtE has qT(x) = h,(x) for x E X , x -A 2 ltT the homotopy type of MC-E~e xl u.. .U e J(') for certain maps ql,. . . ,cpj(c) q,(tu) = tu for oLtLT, u e , .x 'p1 cpj(c) q,(tu) = h2-2t+Tcp(u) for 5 t 5 1, u E e . when E is small enough, and there is a homotopy equivalence h: MC-E - M&. Therefore F a left homotopy inverse. has We have assumed that there is a homotopy equivalence h': M& - K, where K The proof that F is a homotopy equivalence will now be purely is a CW-complex. formal, based on the following. Then each h' 0 h 0 cpj is homotopic by cellular approximation to ASSERTION. If a map F has a left homotopy inverse L and a a map x right homotopy inverse R, then F is a homotopy equivalence; and qj: 6 j - (xj-l) - skeleton of K. R (or L) is a 2-sided homotopy inverse. A Then K u e hl V...u e j(') is a CW-complex, and has the same homotopy PROOF : The relations $1 *j(c)

LF 2 identity, FR 2 identity, tme as M'+", by Lemmas 3.6, 3.7. imply that By induction it follows that each e' has the homotopy type of a LzL(FR) = (LF)R"R. m-complex. If M is compact this completes the proof. If M is not com- Consequently Pact, but all critical points lie in one of the compact sets e, then a RF 2 LF 2 identity , Proof similar to that of Theorem 3.1 shows that the set M& is a deformation

which proves that R is a 2-sided inverse. retract of M, SO the proof is again complete.

The proof of Lemma 3.7 can now be completed as follows. The rela- If there are infinitely many critical points then the above con- tion struction gives us an infinite sequence of homotopy equivalences Ma C Ma2 C M a C ... kGF 2 identity asserts that F has a left homotopy inverse; and a similar proof shows that vvvIll G has a left homotopy inverse. K1 C K2 C K3 C ... ,

Step 1. Since k(GF) 2 identity, and k is known to have a left each extending the previous one. Let K denote the union of the Ki in the inverse, it follows that (GF)k 2 identity. direct limit topology, i.e., the finest possible compatible topology, and 24 I. NON-DEGENERATE FUNCTIONS §4. EXAMPLES 25

let g: M - K be the limit map. Then g induces isomorphisms of homotopy moupg- -in all dimensions. We need only apply Theorem 1 of Combinatorial [Whitehead’s homotopy I to conclude that g is a homotopy equivalence. 94. Examples.

theorem states that if M and K are both dominated by CW-complexes, then As an application of the theorems of $3 we shall prove:

any map M -+ K which induces isomorphisms of homotopy groups is a homotopy THEOREM 4.1 (Reeb). If M is a compact manifold and f equivalence. Certainly K is dominated by itself. To prove that M is is a differentiable function on M with only two critical dominated by a W-complex it is only necessary to consider M as a retract points, both of which are non-degenerate, then M is of tubular neighborhood in some Euclidean space.] This completes the proof homeomorphic to a sphere.

of Theorem 3.5.

REMARK. We have also proved that each has the homotopy type

of a finite CW-complex, with one cell of dimension X for each critical point of index A in Ma. This is true even if a is a critical value.

(Compare Remark 3.4. )

PROOF: This follows from Theorem 3.1 together with the Lemma of Morse (92.2). The two critical points must be the minimum and maximum

points. Say that f(p) = 0 is the mimimum and f(q) = 1 is the maximum.

If e is small enough then the sets ME = f-l[o,~l and f-l[i-E,i] are

closed n-cells by 92.2. But I6 is homeomorphic to M’-€ by 93.1. Thus M is the union of two closed n-cells, M’-€ and f-l[l-~,ll, matched along their common boundary. It is now easy to construct a homeomorphism between M and Sn.

REMARK 1. The theorem remains true even if the critical points are degenerate. However, the proof is more difficult. (Cmpare Milnor, Sommes de varietes differentiables et structures differentiables des sph&res, Bull. SOC. Math. de France,Vol. 87 (19591, pp. 439-444; Theorem l(3); or R. Rosen, A weak form of the star conjecture for manifolds, Abstract 570-28, Notices her. Math Soc.,Vol. 7 (1960), p. 380; Lemma I.)

REMARK 2. It is not true that M must be diffeomorphic to Sn with its usual differentiable structure.(Compare: Milnor, On manifolds homeomor- PNc to the 7-sphere, Annals of Mathematics, Vol. 64 (1956), pp, 399-405. In this paper a 7-sphere with a non-standard differentiable structure is

Proved to be topologically S7 by finding a function on it with two non- J4. EXAMPLES 27 degenerate critical points.) It follows that PO,P,,.. .,Pn are the Only critical points of f. The As another application of the previous theorems we note that if an index of f at pk is equal to twice the number of j with c. < ck. J n-manifold bs a differentiable function on it with only three critical Thus every possible even index between 0 and 2n occurs exactly once. points then they have index 0, n 'and n/2 (by Poincark duality), and the Theorem 3.5: &fold the homotopy type of an n/2-sphere with an n-cell attached. has cpn has the homotopy type of a CW-complex of the form T~Ssituation is studied in forthcoming papers by J. Eel19 and N. Kuiper. e0 u e2 u e4 u..." e2n . Such a function exists example on the real complex projective plane. for or It follows that the integral homology groups of CPn are given by Let CPn be complex projective n-space. We will think of CPn as Z for i = 0,2,4 2n equivalence classes of (n+l)-tuples (z,, ..., zn) of complex numbers, with Hi(CPn;Z) = ,..., 0 for other values of i clzj l2 = 1. Denote the equivalence class of (z,,. . .,zn) by (zo:zl:. . . :zn). Define a real valued function f on CPn by the identity

where cO,cl,..., cn are distinct real constants. In order to determine the critical points of f, consider the

following local coordinate system. Let U, be the set of (zo:zl:... :'n)

Then

are the required coordinate functions, mapping U, diffeomorphically onto the open unit ball in R2". Clearly 2 2 IZjI2 = xj + Yj 1z0l2 = 1 - c (Xj 2 + yj2) so that

throughout the coordinate neighborhood U,. Thus the only critical point of f within U, lies at the center point

p, = (1:o:o: ... :o)

of the coordinate system. At this point f is non-degenerate; and has index equal to twice the number of j with cj < co. Similarly one can consider other coordinate systems centered at the points p, = (0:1:0: ... :o), ...,p, = (0:0:... :0:1) . 95. THE: MORSE INEQUALITIES 28 I. NON-DEGE~TEFUNCTIONS 29 Let M be a compact manifold and f a differentiable function on M with isolated, non-degenerate, critical points. Let a1 (...( ak a Mai i = M. 55. The Morse Inequalities. be such that contains exactly critical points, and M Then a In Morse's original treatment of this subject, Theorem 3.5 was not H,(M iJMai-i) = H,(Mai-i exiMai-i 9 ) The relationship between the topolo@p of M and the critical available. where Xi is the index of the points of a real valued function on M were described instead in terms of critical point, This section will describe this orieinal xi a collection of inequalities. = H,(e ,&Ii) by excision, point of view. coefficient group in dimension Xi DEFINITION: Let S be a function from certain pairs of spaces to = { 0 otherwise.

S is subadditive if whenever X3Y3 2 we have S(X,Z) 5 Applying (1) to 0 = Ma' C.. .C M&k the integers. = M with S = R, we have S(X,y) + S(Y,Z). If equality holds, S is called additive. As an example, given any field F as coefficient group, let

R,(X,Y) = Xth Betti number of (X,Y) where C, denotes the number of critical points of index A. Applying this = rank over F of H,(X,Y;F) , formula to the case S = x we have

for any pair (X,Y) such that this rank is finite. R, is subadditive, as is easily seen by examining the following portion of the exact sequence for

(x,y,Z): Thus we have proven:

. . . - H,(Y,Z) - H,(X,Z) -. %(X,Y) + . . . I

The Euler characteristic X(X,Y) is additive, where X(X,Y) = THEOREM 5.2 (Weak Morse Inequalities). If C, denotes the number of critical points of index h on the compact mani- C (-1)' R,(X,Y). fold M then

(2) R,(M) 5 CX , and LEMMA 5.1. Let S be subadditive and let XoC ...C %. (3) Then S(Xn,Xo)5 2 S(Xi,Xi-l). If S is additive then C (-1)' RX(M) = C (-1)' C, . i=1 equality holds. Slightly sharper inequalities can be proven by the following argument.

PROOF: Induction on n. For n = 1, equality holds trivially and LEMMA 5.3. The function SX is subadditive, where the case n = 2 is the definition of [sub] additivity. n- 1 If the result is true for n - 1, then S(Xn-, ,Xo) 5 c1 S(Xi,Xi-l). s,(x,y) = Rx(X,Y) - RX-,(X,Y) + R,-,(X,Y) - +...? Ro(X,Y) .

PROOF: Given an exact sequence

LALBJ-CL...... + D+O of vector spaces note that the rank of the homomorphism h plus the rank of i is equal to the rank of A. Therefore, NON-DEGENERATE FUNCTIONS 30 I. 95. THE MORSE INEQUALITIES 31 Subtracting this from the equality above we obtain the following: rank h = rank A - rank i

= rank A - rank B + rank j COROWY 5.4. If Ch+l = C,-l = 0 then R, = C, and = rank A - rank B + rank C - rank k ... Rb+l = R,-l = 0.

= rank A - rank B + rank C - +...+ rank D . Now consider the homoiogy exact sequence (Of course this would also follow from Theorem 3.5.) Hence the last expression is 0. Note that 2 this corollary enables us to find the homology .goups of complex projective of a triple X 2 Y 1 2. Applying this computation to the homomorphism space (see 94) without making use of Theorem 3.5. Hx+l(X,Y) a- %(Y,Z)

we see that

rank a = R,(Y,Z) - R,(X,Z) + R,(X,Y) - R,-l(Y,Z) + ... 2 0

Collecting terms, this means that

which completes the ?roof.

Applying this subadditive function S, to the spaces a &k 0 C Ma1 C M2 C...C M we obtain the Morse inequalities

( 4,) R,(M) - R,-l (M) +-. . .+ Ro(M) 5 C, - C,-l + -. .+ C,. These inequalities are definitely sharper than the previous ones.

In fact, adding (4,) and (4,-1), one obtains (Zh); and comparing (4,) with (4,-,) for A > n one obtains the equality (3). As an illustration of the use of the Morse inequalities, suppose

that Cx+l = 0. Then RX+l must also be zero. Comparing the inequalities (4,) and (4x+1), we see that

R, - R,-l +-...+ R0 = C, - C,-l +-...t C, . I

Now suppose that C,-l is also zero. Then R,-l = 0, arid a similar argument shows that

Rh-2 - R,-3 +-...+_ R, = Ch-2 - C,-3 +-...+ Co . , I. NON-DEGENERATE FUNCTIONS §6. MANIFOLDS IN ETJCLIDM SPACE 32 33

THEOREM 6.1 (Sard). If M, and M2 are differentiable manifolds having a countable basis, of the same dimension, and f: M1 - M, is of class C1, then the image of the 56. Manifolds in Euclidean Space. set of critical points has measure 0 in M,.

Although we have so far considered, on a manifold, only functions A critical point of f is a point where the Jacobian of f is which have no degenerate critical points, we have not yet even shown that singular. For a proof see de Rham, “Vari6tBs Differentiables, ‘I Hermann, such functions always exist. In this section we will construct many func- Paris, 1955, p. 10. tions with no degenerate critical points, on any manifold embedded in Rn. COROLLARY 6.2. For almost all x E Rn, the point x is In fact, if for fixed p cRn define the function 5:M -R by $(q) = not a focal point of M. p, the function 5 has ((p-q1I2. It will turn out that for almost all only non-degenerate critical points. PROOF: We have just seen that N is an n-manifold. The point x Let M C Rn be a manifold of dimension k < n, differentiably em- is a focal point iff x is in the hge of the set of critical points of Let N C M x Rn be defined by bedded in R”. E: N -R”. Therefore the set of focal points measure has 0. N = [(q,v): q E M, v perpendicular to M at 9). For a better understanding of the concept of focal point, it is con- N is an n-dimensional manifold It is not difficult to show that venient to introduce the “second fundamental form“ of a manifold in Euclidean (N is the total space of the normal differentiably embedded in R2n. space. We will not attempt to give an invariant definition; but will make vector bundle of M.) use of a fixed local coordinate system. (E is the “endpoint“ mSp.) Let N+Rn be E(q,v) = q + V. E: Let ul,. . . ,uk be coordinates for a region of the &fold M C Rn. Then the inclusion map from M to Rn determines n smooth functions

x,(u1 ,...,u k ) ,..., x,(u 1 )...)u k ) .

These functions will be written briefly as z(u’, . . .,uk) -L where x = (xl,. . . ,%) . To be consistent the point q E M C Rn will now be denoted by -L 9. The first fundamental form associated with the coordinate system is defined to be the symmetric matrix of real valued functions

DEFINITION. e E Rn is a focal point of (M,q) with multiplicity

p if e = q + v where (q,v) E N and the Jacobian of E at (4,v) has The second fundamental form on the other hand, is a symmetric matrix if Fij) nullity p > 0. The point e will be called a focal point of M e is of vector valued functions. a focal point of (M,q) for some q E M. It is defined as follows. The vector a% at a point of M can Intuitively, a focal point of M is a point in Rn where nearby be expressed as the sum of a vector tangent to MaU andau a vector normal to M.

normals intersect. Define yij to be the normal component of 3% . Given any unit vector -c au We will use the following theorem, which we will not Prove. v which is normal to M at the matrix aU 86. MANIFOLDS IN EUCLIDEAN SPACE I. NON-DEGENERATE FUNCTIONS 35 34

a? + + vectors -aul'"'l auk1 a; w1 9'*' ,wn-k we will obtain an nxn matrix whose can be called the "second fundamental form of M at z in the direction d I1 rank equals the rank of the Jacobian of V. E at the corresponding point. It will simplify the discussion to assume that the Coordinates This nxn matrix clearly has the following form -+ mve been chosen to that g.ljr evaluated at 9, is the identity matrix. men the eigenvalues of the matrix ( 3 . dij) are called the principal ,matures K1,.. .,Kn of M at 3 in the normal direction 7. The re- ,iprocals K;~,. . . of these principal curvatures are called the princi- \ 0 identity pi radii of curvature. Of course it may happen that the matrix (7 . dij) matrix will be zero; and hence is singular. In this case one or more of the Thus the rank is equal to the rank of the upper left hand block . Using the corresponding radii K;' will not be defined. the identity Now consider the normal line 1 consisting of all z + t?, where + v is a fixed unit vector orthogonal to M at , we see that this upper left hand block is just the matrix LEbMA 6.3. The focal points of (M,<) along P are pre- + cisely the points q + K;' 3, where 1 5 i 5 k, $ # 0. Thus there are at most, k focal points of (M,;) along P, each being counted with its proper multiplicity. Thus : +1 k + 1 k ASSERTION 6.4. PROOF: Choose n-k vector fields wl(u ,..., u ),...,Wn-k(u ,...,U ) + t; is a focal point of (M,a with multiplicity -+ -+ along the manifold so that w,, ...jWn-k are unit vectors which are orthogo- 1.1 if and only if the matrix 1 k nal to each other and to M. We can introduce coordinates (U ,...,U , 4 (*) 'glj - tT; . -Ti$ 1 k1 t', ...,tn-k) on the manifold N C M x Rn as follows. Let (U ,...,U ,t ,..., is singular, with nullity 1.1. tn-k) correspond to the point Now suppose that (g ) is the identity matrix. Then (+) is sin&- nf tQ;a(ul,...,u k )) E N . lar If and only if (Z(u1 , ... ,u ) , is an eigenvalueij of the matrix (7 . Ti,) . Further- a=1 more the multiplicity IJ is equal to the multiplicity of $ as eigenvalue. This completes the proof Lemma 6.3. Then the function of E: N +Rn Now for fixed 'i; E Rn let us study the function gives rise to the correspondence 3 = f: M +R where

2; * + . T; . with partial derivatives 5 i; We have I. NON-DEGENERATE FUNCTIONS §6. MANIFOLDS IN EXJCLIDEAN SPACE 37

The second partial derivatives at a critical point are given by It follows that the sum of the indices of any vector field on M is equal to X(M) because this sum is a topological invariant (see Steen- rod, "The Topology of Fibre Bundles," 139.7). -- The preceding corollary can be sharpened as follows. Setting 3 = x + tv, as in the proof of Lemma 6.3, this becomes Let k 2 o be an integer and let K C M be a compact set.

COROLLARY 6.8. Any bounded smooth function f: M-R can

Theref Ore: be uniformly approximated by a smooth function g which has no degenerate critical points. Furthermore g can be LEMMA 6.5. The point E M is a degenerate critical point chosen so that the i-th derivatives of g on the compact of f=Lp if and only if is a focal point of (M,T). set K uniformly approximate the corresponding derivatives The nullity of as critical point is equal to the multi- of f, for i 5 + k. plicity of p as focal point. (Compare M. Morse, The critical points of a function of n vari- Combining this result with Corollary 6.2 to SarG's theorem, we ables, Transactions of the American Mathematical Society, Vol. 33 (1931), immediately obtain: pp. 71-91 .)

THEOREM 6.6. For almost all p E Rn (all but a set of PROOF: Choose some imbedding h: M -.Rn of M as a bounded sub- measure 0) the function set of some euclidean space so that the first coordinate h, Lp: M-R is precisely the given function f. Let c be a large number. Choose a point has no degenerate critical points. p = (-C+E1 ,E2,. . .,En) This theorem has several interesting consequences. close to (-c,O,. ..,O) E Rn so that the function : M - R is non- COROWY 6.7. On any manifold M there exists a dif- degenerate; and set L.p ferentiable function, with no degenerate critical points, - - c2 for which each Ma is compact. g(x) - 2c Clearly g is non-degenerate. A short computation shows that PROOF: This follows from Theorem 6.6 and the fact that an n-dimensional manifold M can be embedded differentiably as.&closed subset of R2n+1 (see Whitney, Geometric Integration Theory, p. 113). Clearly, if c is large and the Ei are small, then g will approximate APPLICATION 1. A differentiable manifold has the homotopy type of f as required. a CW-complex. This follars from the above corollary and Theorem 3.5. The above theory can also be used to describe the index of the function APPLICATION 2. On a compact manifold M there is a vector field X such that the sum of the indices of the critical points of X equals X(M), the Euler characteristic of M. This can be seen as follows: for at a critical point any dlfferentiable function f on M we have x(M) = C (-1)' C, where C, LEMMA 6.9. (Index theorem for Lp. The index of is the number of critical points with index A. But (-1)' is the index of 5 at a non-degenerate critical point q E M is equal to the vector field grad f at a point where f has index the number of focal points of (M,q) which lie on the sepent from q to p; each focal point being counted with its multiplicity. 57. THE LEFSCHETZ T€EOREM 39 I. NON-DEGENERATE FUNCTIONS 38

An analogous statement in Part I11 (the Morse Index Theorem) will be of fundamental importance. $7. The Lefschetz Theorem on Hyperplane Sections. PROOF: The index of the matrix As an application of the.ideas which have been developed, we will prove some results concerning the topology of algebraic varieties. These

Assuming that (gij) is were originally proved by Lefschetz, using quite different arguments. The is equal to the number of negative eigenvalues. * present version is due to Andreotti and Frankel . the identity matrix, this is equal to the number of eigenvalues of (7 bij) 1 are 2 . Comparing this statement with 6.3, the conclusion follows. THEOREM 7.1. If M c C" is a non-singular affine algebraic variety in complex n-space with real dimension 2k, then Hi(M;Z) = 0 for i > k.

This is a consequence of the stronger:

THEOREM 7.?. A complex analytic manifold M of complex dimension k, bianalytically embedded as a closed subset of cn has the homotopy type of a k-dimensional CW-complex.

The proof will be broken up into several steps. First consider a quadratic form in k complex variables

Q(z1 ,..., k ) = lbhj zhzj .

If we substitute xh + iyh for zh, and then take the real part of Q we obtain a real quadratic form in 2k real variables:

Q'(x', ...,xk,y', ...,y k ) = real part of Tbhj(xh+iyh)(xd+iyj).-. .

ASSERTION 1. If e is an eigenvalue of Q' with multiplicity v,

then -e is also an eigenvalue with the same multiplicity p.

PROOF. The identity Q( iz' , . . . ,izk ) = -&( z1 , . . . ,zk) shows that the quadratic form &' can be transformed into -&I by an orthogonal change of variables. Assertion 1 clearly follows.

* See S. Lefschetz, "L'analysis situs et la ggornetrie alg6briaue."- Paris

1924; and A. Andreotti and T. Frankel, The Lefschetz theorem on hyperplane-7 sections, Annals of Mathematics, Vol. 69 '( 1959), pp. 71 3-71 7. 17. THE LEFSCHETZ THEOREM I. NON-DEGENERATE F7JNCTIONS 41 40

Now consider a complex manifold M which is bianalytically imbed- $: M-R ded as a subset of Cn. Let q be a point of M. has no degenerate critical points. Since M is a closed subset of Cn, it is clear that each set ASSERTION 2. The focal points of (M,q) along any normal line fi I@ = $‘[o,al in pairs which are situated symmetrically about q. is compact. Now consider the index of $ at a critical point q. Accord- ing to 6.9, this index is equal to the number focal points of In other words if q + tv is a focal point, then q - tv is a of (M,q) focal point with the same multiplicity. which lie on the line segment from p to q. But there are at most 2k focal points along the full line through p and q; and these are distri- 1 PROOF. Choose complex coordinates z , . . . ,zk for M in a neigh- buted symmetrically about q. Hence at most k of them can lie between p 1 The inclusion map M + Cn borhood of q so that z (4) = . . . = zk(q) = 0. and q. determines n complex analytic functions Thus the index of $ at q is 5 k. It follows that M has the 1 k w, = w,(z ,..., z ), a = 1 ,..., n. homotopy type of a CW-complex of dimension 2 k; which completes the proof of 7.2. Let v be a fixed unit vector which is orthogonal to M at q. Consider

the Hermitian inner product COROLLARY 7.3 (Lefschetz). Let V be an algebraic variety of complex dimension k which lies in the complex projective space CP,. Let P be a hyperplane in CPn which contains of w and v. This can be expanded as a complex power series the singular points (ifany) of V. Then the inclusion map Vn P-+V lw,(zl ,..., zk)C, = constant + Q(z’, ...,zk) + higher terms, induces isomorphisms of homology groups in dimensions less where Q denotes a homogeneous quadratic function. (The linear terms van- than k-1. Furthermore, the induced homomorphism ish since v is orthogonal to M.) H~-,(v n P;Z) - H~-~(v;z) Now substitute xh + iyh for zh so as to obtain a real coordinate is onto. system for M; and consider the real inner product PROOF. Using the exact sequence of the pair (V,V n P) it is w * v = real part of Wac, . 1 clearly sufficient to show that q(V,V n P;Z) = 0 for r 5 k-1. But the This function has the real power series expansion Lefschetz duality theorem asserts that k1 w * v = constant + ~‘(x’,, ,x ,y , . . ,yk) + higher terms. . . . q(v,v n P;Z) = H~~-~(v-(v n P) ;z) . Clearly the quadratic terms Q’ determine the second fundamental form of But V -(V n p) is a non-singular algebraic variety in the affine space CPn - P. M at q in the normal direction v. By Assertion 1 the eigenvalues of Hence it follows from 7.2 that the last group is zero for r 5 k-1. Q’ occur in equal and opposite pairs. Hence the focal points of (M,q) This result can be sharpened as follows: along the line through q and q + v also occur in symnetric pairs. This THEOREM 7.4 (Lefschetz). Under the hypothesis of the proves Assertion 2. preceding corollary, the relative homotopy group € We are now ready to prove 7.2. * Choose a point p Cn SO that the a,(V,V n P) is zero for r < k. squared-distance function I. NON-DEGENERATE FUNCTIONS 42 43

PROOF. The proof will be based on the hypothesis that Sme neighborhood U of V n P can be deformed into V n P within V. hi^ can be proved, for example, using the theorem that algebraic varieties can be tri- awlated. In place of the function Lp : V - V n P -c R we will use f: V --c R where ro for x EV~P, PART I1 f(x) = i I/L~(X) for x B p. A RAPID COURSE IN RIEMANNIAN GEOMETRY Since the critical points of Lp have index -< k it follows that the critical points of f have index 2 2k - k = k. The function f has §8. Covariant Differentiation no degenerate critical points with E 5 f < m. Therefore V has the 1 The object of Part I1 will be to give a rapid outline of some basic homotopy type of VE = f- [O,E~ with finitely many cells of dimension 2 k attached. concepts of Riemannian geometry which will be needed later. For more infor- that VE C U. Let 1' denote the unit mation the reader should consult Nomizu, "Lie groups and differential Choose E small enough so (Ir,fr) into (V,V n P) can be deform- geometry," The Mathematical Society of Japan, 1956; Laugwitz, "Differential- r-cube. Then every map of the pair geometrie," Teubner 1960; or Helgason, "Differential geometry and symmetric ed into a map spaces," Academic Press, 196?. (IF,F) - (VE,V n P) C (u,V n P) , Let M be a smooth manifold. since r < k, and hence can be deformed into V n P. This completes the DEFINITION. An affine connection at a point p E M is a function proof. which assigns to each tangent vector Xp E T% and to each vector field Y a new tangent vector

Xp k Y E T%

called the convariant derivative* of Y in the direction X This is re- P' quired to be bilinear as a function of X and Y. Furthermore, if P f: M-R

is a real valued function, and if fl denotes the vector field

(fYIq = then k is required to satisfy the identity

* Note that our X b Y coincides with Nomizu's ",Y. The notation is in- tended to suggest that the different:al operator X acts on the vector field Y. 44 11. RIE”IAN GEOMETRY 8 8. COVARIANT DIFFERENTIATION 45

where the symbol yk stands for the real valued function (As usual, %f denotes the directional derivative of f in the direction ,i

X P -I A global affine connection (or briefly a connection) on M is a

function which assigns to each p E M an affine connection kp at P, rijk on satisfying the following smoothness condition. Conversely, given any smooth real valued functions u,

1) If X and Y are smooth vector fields on M then the vector one can define X k Y by the formula (6). The result clearly satisfies

field X I- Y, defined by the identity the conditions (11, (21, (3), (4), (5). Using the connection I- one can define the covariant derivative of (X I- Y), = xp kp Y , a vector field along a curve in M. First some definitions. must also be smooth. A parametrized curve in M is a smooth function c from the real Note that: numbers to M. A vector field V along the curve c is a function which (2) X I- Y is bilinear as a function of X and Y . assigns to each t E R a tangent vector (3) (fx) I-Y = f(X I- Y) , (4) (X I- (fY) = (Xf)Y + f(X kY) . Vt TMc(t) . Conditions (1) , (2) , (3) , (4) can be taken as the definition of This is required to be smooth in the following sense: For any smooth func- a connection. tion f on M the correspondence 1 In terms of local coordinates u ,. . . ,un defined on a coordinate t -Vtf neighborhood U C M, the connection I- is determined by n3 smooth real k should define a smooth function on R. valued functions rij on U, as follows. Let ak denote the vector As an example the velocity vector field $ of the curve is the field a on U. Then any vector field X on U can be expressed 2 vector field along c which is defined by the rule uniquely as xdc = c*xd

k=1 d Here denotes the standard vector field on the real numbers, and where the xk are real valued functions on U. In particular the vector field a, k aj can be expressed as denotes the homomorphism of tangent spaces induced by the map c. (Compare Diagram 9.)

These functions rki3 determine the connection completely on U. In fact given vector fields X = lxiai and Y = L$aj one can

expand X I- Y by the rules (2), (3) , (4) ; yielding the formula

Diagram g 46 11. RIEMANNIAN GEOMETRY § 8. COVARIANT DIFFERENTIATION 47

Now suppose that M is provided with an affine connection. Then LEMMA 8.2. Given a curve c and a tangent vector V, DV at the point c(o), there is one and only one parallel any vector field V along c determines a new vector field along c vector field V along c which extends Vo. called the covariant derivative of V. The operation DV v-Z€ PROOF. The differential equations

is characterized by the following three axioms. D(V+W) DV DW a) = =+=. have solutions vk(t) which are uniquely determined by the initial values b) If f is a smooth real valued function on R then DV vk(0). Since these equations are linear, the solutions can be defined for atD(fV) = $V+f= . all relevant values of t. (Compare Graves, "The Theory of Functions of c) If V is induced by a vector field Y on M, that is if Real Variables," p. 152.) dc for each t, then is equal to -m k Y Vt = Yc(t) The vector Vt is said to be obtained from Vo by parallel trans- in the direction of the (= the covariant derivative of Y lation along c. velocity vector of c) Now suppose that M is a Riemannian manifold. The inner product DV LFDlMA 0.1 . There is one and only one operation V - -m of two vectors Xp, Yp will be denoted by . which satisfies these three conditions. DEFINITION. A connection 1 on M is compatible with the Rieman- PROOF: Choose a local coordinate system for M, and let nian metric if parallel translation preserves inner products. In other words, ul(t),. .,un(t) denote the coordinates of the point c(t). The vector for any parametrized curve c and any pair P, P' of' parallel vector fields field V can be expressed uniquely in the form along c, the inner product should be constant.

LEMMA 8.3. Suppose that the connection is compatible with where v' , . . . ,vn are real valued functions on R (or an appropriate open the metric. Let V, W be any two vector fields along c. Then subset of R), and a,, ...,an are the standard vector fields on the coordinate neighborhood. It follows from (a), (b), and (c) that

PROOF: Choose parallel vector fields P,, ...,Pn along c which are orthonormal at one point of c and hence at every point of c. Then the given fields V and W can be expressed as 1 viPi and 2 wjPj respectively (where vi = is a real valued function on R). It follows that = 1 viwi and that Conversely, defining w by this formula, it is not difficult to verify

that conditions (a), (b), and (c) are satisfied. A vector field V along c is said to be a parallel vector field DV Theref ore if the covariant derivative is identically zero.

which completes the proof. § 8. COVARIANT DIFFERENTIATION 49 48 11. RIEMANNIAN GEOMETRY

three quantities COROLLARY 8.4. For any vector fields Y,Y' on M and any vector Xp E TMp: I- aj,ak>, ,and * xp = + . (There are only three such quantities since a, t aj = aj 1 a, .) These equations can be solved uniquely; yielding the first Christoffel identity PROOF. Choose a curve c whose velocity vector at t = 0 is X P; ' aj,ak> = 71 (aigjk + aj@;ik- a@ij) and apply 8.3. ' The left hand side of this identity is equal to rijP gQk . Multiplying DEFINITION 8.5. A connection t is called symmetric if it satis- 1 P fies the identity* by the inverse (gkP) of the matrix (gPk) this yields the second Christof- (X t Y) - (Y I- X) = [X,YI I fel identity

(AS usual, [X,YI denotes the poison bracket [X,Ylf = X(Yf) - Y(Xf) of two vector fields.) Applying this identity to the case X = a,, Y = a j' since [a13 ,a 1 = 0 one obtains the relation Thus the connection is uniquely determined by the metric. k k rij - rji = 0. Conversely, defining rijP by this formula, one can verify that the resulting connection is symmetric and compatible with the metric. This k then using formula (6) it is not difficult to Conversely if rij = rji completes the proof. verify that the connection k is symmetric throughout the coordinate neigh- An alternative characterization of symmetry will be very useful borhood. later. Consider a "parametrized surface'' in M: that is a smooth function

LEMMA 8.6. (Fundamental lemma of Riemannian geometry.) s: R2 -c M . A Riemnnian manifold possesses one and only one symmetric connection which is compatible with its metric. By a vector field V along s is meant a function which assigns to each (x,y) 6 R* a tangent vector (Compare Nomizu p. 76, Laugwitz p. 95.) YX,Y) TMS(x,Y) . PROOF of uniqueness. Applying 8.4 to the vector fields ai,aj,a,, As examples, the two standard vector fields a and & give rise to vec- and setting < > = gjk one obtains the identity tor fields s, a and s* a along s. These will be denoted briefly by a j' a k as and $ ; and called the "velocity vector fields" of s. a, gJk = < a, t aj,ak> + < aj,ai t ak> . For any smooth vector field V along s the covariant derivatives Permuting i,j, and k this gives three linear equations relating the % and 5DV are new vector fiedds, constructed as follows. For each fixed * yo, restricting V to the curve The following reformulation may (or may not) seem more intuitive. Define The "covariant second derivative" of a real valued function f along two x - S(X,Y0) vectors $,Yp to be the expression one obtains a vector field along this curve. Its covariant derivative with xp(Yf) - '5 k Y)f respect to x is defined to be )(x,y,) . This defines % along where Y denotes any vector field extending Y It can be verified that ( P' the entire parametrized surface s. this expression does not depend on the choice of Y. (Compare the proof of LRmma 9.1 below.) Then the connection is symmetric if this second deriva- As examples, we can form the two covariant derivatives of the two tive is symmetric as a function of and Y Xp P' 11. RIEMANNIAN GEOMETRY 50 §9. THE CURVATURE TENSOR 51 as as D as D as vector fields and . The derivatives and are simply the acceleration vectors of suitable coordinate curves. However, the mixed derivatives as and cannot be described so simply. §9. The Curvature Tensor X5 -& The curvature tensor R of an affine connection t measures the D as D as extent to which the second covariant derivative a, t- (a. t Z) is sym- LEMMA 8.7. If the connection is symmetric then -ax -ay = --ay ax . J metric in i and j. Given vector fields X,Y,Z define a new vector field * R(X,Y)Z by the identity PROOF. Express both sides in terms of a local coordinate system, R(X,Y)Z = -X t- (Y 1 Z) + Y k (X t- Z) + [X,YI t Z . and compute.

LEMMA 9.1. The value of R(X,Y)Z at a point p E M depends only on the vectors X ,Y ,Z at this point PPP p and not on their values at nearby points. Further- more the correspondence XPJPJP - R(XpJp)Zp from T$ x T% x T$ to T$ is tri-linear.

Briefly, this lema can be expressed by saying that R is a "tensor."

PROOF: Clearly R(X,Y)Z is a tri-linear function of X,Y, and Z. If X is replaced by a multiple fX then the three terms -X t- (Y I- Z), Y 1 (X t- Z), [X,YI t- 2 are replaced respectively by i) -fX t-(Y t-Z) , ii) (Yf)(X t- Z) + fY 1 (X t- Z) , iii) - (Yf)(X t- Z) + f[X,YI t- z . Adding these three terms one obtains the identity

R(fX,Y)Z = fR(X,Y)Z .

Corresponding identities for Y and Z are easily obtained by similar computations. Now suppose that X = 1 Y = 1 Yiaj , and z = 1 zkak. Then R(X,Y)Z = z R(xiai,yjaj) (zkak)

* Nomizu gives R the opposite sign. Our sign convention has the advan- tage that (in the Riemannian case) the inner product coincides with the classical symbol R hijk ' § 9. TB CURVATURE TENSOR 53 52 11. RIEMANNIAN GM)METRY D = 0, this implies that & P is identically zero; Evaluating this expression at p one obtains the formula Since ( 3% P)(Xo,O) and completes the proof that P is parallel along 3.1 (R(X,Y)Z)p = X'(P)Yj(P)zk(P) (R(ai,aj)ak\, 1 Henceforth we will assume that M is a Riemannian manifold, pro- WMch depends only on the values of the functions xi,yj,zk at p, and vided with the unique symmetric connection which is compatible with its not on their values at nearby points. This completes the proof. metric. In conclusion we will prove that the tensor R satisfies four Now consider a parametrized surface symmetry relations.

s: R2-M. LEMMA 9.3. The curvature tensor of a Riemannian manifold Given any vector field V along s. one can apply the two covariant dif- satisfies: D ferentiation operators and 3 to V. In general these operators will (1) R(X,Y)Z + R(Y,X)Z = 0 not commute with each other. (2) R(X,Y)Z + R(Y,Z)X + R(2,X)Y = 0 (3) + = 0 DD DD as as umim9.2. *xv-x*v = R(Z'*b . (4) = .

PROOF: Express both sides in terms of a local coordinate system, PROOF: The skew-symmetry relation (1) follows immediately from the and compute, making use of the identity definition of R. aj I- (ai F a,) - a, t- (aj F a,) = R(ai,aj)ak . Since all three terms of (2) are tensors, it is sufficient to

[Itisinteresting to ask whether one can construct a vector field prove (2) when the bracket products [X,YI, [X,Zl and [Y,Zl are all P along s which is parallel, in the sense that zero. Under this hypothesis we must verify the identity D D zp= EP = 0, -x k(Y I-2) + Y I-(X t-2) and which has a given value p(o,o) at the origin. In general no such -Y I-(2 I-X) + z F(Y I-X) vector field exists. However, if the curvature tensor happens to be zero - 2 I-(X I-Y) + x I-(2 I-Y) = 0.

then P can be constructed as follows. Let P(x,o) be a parallel vector But the symmetry of the connection implies that field along the x-axis, satisfying the given initial condition. For each Y 1 z - z kY = [Y,ZI = 0 . fixed xo let P be a parallel vector field along the curve (Xo,Y) Thus the upper left term cancels the lower right term. Similarly the re- Y - S(X0,Y) , maining terms cancel in pairs. This proves (2). bving the right value for y = 0. This defines P everywhere along s. To prove (3) we must show that the expression is Clearly P is identically zero; and P is zero along the x-axis. Y & skew-symmetric in 2 and W. This is clearly equivalent to the assertion Now the identity that DD DD 3xP-axyP = R(,%&% = 0 = O DD D implies that *x P = 0. In other words, the vector field P is for all X,Y,Z. Again we may assume that [X,Yl = 0, SO that parallel along the curves is equal to

Y - S(X0,Y) . <- x I- (Y I- 2) + Y t- (X t- Z),Z> . 54 11. RIEMANNIAN GEOMETRY § 10. GEODESICS AND COMPLETENESS 55

In other words we must prove that the expression $10. Geodesics and Completeness is symmetric in X and Y. Let M be a connected Riemannian manifold. Since [X,YI = 0 the expression YX is symmetric in X DEFINITION. A parametrized path and Y. Since the connection is compatible with the metric, we have y: I -+M, = 2 1 > x (x z,z where I denotes any interval of real numbers, is called a geodesic if the hence acceleration vector field & $ is identically zero. Thus the velocity Yx = 2 (Y I-(X k Z),Z> + 2 (x k Z,Y I- z> . vector field 8 must be parallel along y. If 7 is a geodesic, then the But the right hand term is clearly symmetric in X and Y. Therefore identity (Y k (X I- Z) ,Z> is symmetric in X and Y; which proves property (3).

Property (4) may be proved from (I), (21, and (3) as follows. shows that the length Il$ll = <%, $>”’ of the velocity vector is constant along 7. Introducing the arc-length function

s(t) = 1 @lldt + constant

This statement can be rephrased as follows: The parameter t along a geodesic is a linear function of the arc-length. The parameter t is actually equal to the arc-length if and only if lldyIIaT = 1. In terms of a local coordinate system with coordinates u 1 , . . . ,u“ 1 a curve t - 7(t) E M determines n smooth functions u (t) , . . . ,un(t). The equation & 8 for a geodesic then takes the form

Formula (2) asserts that the sum of the quantities at the vertices of shaded triangle W is zero. Similarly (making use of (1) and (3)) the sum of the vertices of each of the other shaded triangles is zero. Adding The existence of geodesics depends, therefore, on the solutions of a certain these identities for the top two shaded triangles, and subtracting the system of second order differential equations. identities for the bottom ones, this means that twice the top vertex minus More generally consider any system of equations of the form twice the bottom vertex is zero. This proves (4), and completes the proof. d2;’ = T(<,-df;d< ) . dt2 + Here u stands for (u’, ..., un) and 7 stands for an n-tuple of Cm functions, all defined throughout some neighborhood U of a point

(zl,Tl) E R2” . 11. RIEWW"NN GEOMETRY § 10. GEODESICS AND COMPLETENESS 57

MISTENCE AND UNIQUENESS THEOREM 10.1. There exists a numbers E~,E~> 0 so that: for each p E U and each v E TMp with neighborhood W of the point (T, ,Yl) and a number I1vII < E~ there is a unique geodesic E > 0 so that, for each (To,To)E W the differen- yv: (-2~~,2~~)-+M

satisfying the required initial conditions. has a unique solution t -<(t) which is defined for To obtain the sharper statement it is only necessary to observe that It\ < E, and satisfies the initial conditions the differential equation for geodesics has the following homogeneity property. Let c be any constant. If the parametrized curve Furthermore, the solution depends smoothly on the initial conditions. In other words, the correspondence is a geodesic, then the parametrized curve Go,Yo, t) + 3 t)

from x (-E,E) to R~ is a function of all w ern t -+ Y(Ct) 2n+1 variables. will also be a geodesic. i dui PROOF: Introducing the new variables v = this system of n Now suppose that E is smaller than Then if I/v((< E and second order equations becomes a system of ?n first order equations: (tl < 2 note that

I(v/E~II < and (E2t( <2~~.

Hence we can define yv(t) to be yVlE2(~,t) . This proves lo.?.

This following notation will be convenient. Let v E TM be a The assertion then follows from Graves, "Theory of Functions of Real Vari- 9 tangent vector, and suppose that there exists a geodesic ables," p. 166. (Compare our 52.4.) Applying this theorem to the differential equation for geodesics, y: [O,ll -+M one obtains the following. satisfying the conditions

Y(0) = 9, ,(o)dy = v. LEMMA 10.2. For every point po on a Riemannian manifold M there exists a neighborhood U of p, Then the point y(1) E M will be denoted by exp (v) and called the 4 and a number E > 0 so that: for each p E U and * exponential of the tangent vector v. The geodesic y can thus be des- each tangent vector v E T% with length < E there is a unique geodesic cribed by the formula

yV: (-2,2) +M r(t) = expq(tv) . satisfying the conditions

dYV YV(O) = P, (0) = v . * The historical motivation for this terminology is the following. If M is the ~oupof all n x n unitary matrices then the tangent space TMI at the identity can be identified with the space of n x n skew-Hermitian PROOF. If we were willing to replace the interval (-2,p) by an matrices. The function

arbitrarily small interval, then this statement would follow immediately expI: TMI -+ M from 10.1. To be more precise; there exists a neighborhood U of Po and as defined above is then given by the exponential power series

expI(A) = I + A + -&A2 + &A3 + ... . 11. RIEMANNIAN GEOMETRY 58 §lo. GEODESICS AND COMPLETENESS 59

of (p,p) E M x M. We may assume that the first neighborhood V' consists -ma 10.2 says tbt expq(v) is defined proving that IIvi/ is small enough. of all pairs (q,v) such that q belongs to a given neighborhood U' of 1n general, exp (v) is not defined for large vectors v. However, if q p and such that //v//< E. Choose a smaller neighborhood W of p so that defined at all, expq(v) is always uniquely defined. F(V1) 1 W x W. Then we have proven the following. DEFINITION. The manifold M is geodesically complete if expq (v)

is defined for all 9 E M and all vectors v E T%. This is clearly equiva- LEMMA 10.3. For each p E M there exists a neighborhood lent to the following requirement: W and a number E > 0 so that: (1) Any two points of W are joined by a unique For every geodesic segment yo: [a,bl -M it should be possible geodesic in M of length < E. to extend To to an infinite geodesic (2) This geodesic depends smoothly upon the two y: R +M . points. (I.e., if t-* expql(tv), 0 5 t 5 1, is the geodesic joining q1 and q,, then the pair (ql,v) E We will return to a study of completeness after proving some local results. TM depends differentiably on (ql ,q2) .) Let TM be the tangent manifold of M, consisting of all pairs each q E W the map expq maps the open diffeomorphically onto an open set (p,v) with p E M, v E T%. We give TM the following C" structure: us 1 w. if (u' , . . . ,un) is a coordinate system in an open set U C M then every REMARK. With more care it would be possible to choose W so that tangent vector at q E U can be expressed uniquely as t'a, +. . .+ tnan, a the geodesic joining any two of its points lies completely within where a, = . Then the functions u1 ,...,un,t' ,..., tn constitute W. Com- '9 a coordinate systems on the open set TU C TM. pare J. H. C. Whitehead, Convex regions in the geometry of paths, Quarter- (1932), Lemma 10.2 says that for each p E M the map ly Journal of Mathematics (Oxford) Vol. 3, pp. 33-42. Now let us study the relationship between geodesics and arc-length. (q,v) - expq(v)

is defined throughout a neighborhood V of the point (p,O) E TM. Further- THEOREM 10.4. Let W and E be as in Lemma 10.3. Let more this map is differentiable throughout V. y: [0,11 -* M Now consider the smooth function F: V M x M defined by - be the geodesic of length < E joining two points of W, F(q,v) = (q, exp,(v)). We claim that the Jacobian of F at the point and let o: [0,11 -* M (P,O) is non-singular. In fact, denoting the induced coordinates on 1 be any other piecewise smooth path joining the same two x U C M x M by we have U (d,.. .,u~,u2,..., u:), points. Then 1 Il$lldt 5 s Il$lldt , 0 0 --)a = a F*( atJ -a$ where equality can hold only if the point set c~([o,il) coincides with y( [0,1 1).

Thus the Jacobian matrix of F at (p,O) has the form Thus y is the shortest path joining its end points. hence is non-singular. The proof will be based on two lermnas. It follows fromthe implicit function theorem that F mPS Some Let q = y(o) and let Us be as in 10.3. neighborhood V' of (p,o) E TM diffeomorphically onto some neighborhood §TO. GEODESICS AND COMPLETENESS 61 60 11. RIEMANNIAN GEOMETRY Thus the shortest path joining two concentric spherical shells LEMMA 10.5. In Uq, the geodesics twough q are the orthogonal trajectories of hypersurfaces around q is a radial geodesic.

expq(v) : v TMq, 11v11= constant . PROOF: Let f(r,t) = exp (rv(t)), so that w(t) = f(r(t),t) { E 1 q Then PROOF: Let t- v(t) denote any curve in TMq with Ilv(t) I( = 1,

We must show that the corresponding curves Since the two vectors on the right are mutually orthogonal, and since t - exPq(rov(t) ) llaFllar = I, this gives

in us, where 0 < ro < E, are orthogonal to the radial geodesics r - exp9 (rv(tO)) . where equality holds only if af = 0; hence only if $ = 0. Thus In terms of the parametrized surface f given by

f(r,t) = exp,(rv(t)), o 5 r < E , we must prove that a a af af Zlat; = < > 0 where equality holds only if r(t) is monotone and v(t) is constant. for all (r,t). This completes the proof. Now a ar af D ar ar af D af The proof of Theorem 10.4 is now straightforward. Consider any

3-F = + piecewise smooth path w from q to a point The first expression on the right is zero since the curves q' = exp (rv) E Us ; r -, f(r,t) 9 where 0 < r < E, llvll = 1. Then for any 6 > 0 the path w mist con- are geodesics. The second expression is equal to tain a segment joining the spherical shell of radius 6 to the spherical shell of radius r, and lying between these two shells. The length of this af ar ar segment will be 2 r - 6; hence letting 6 tend to 0 the length of m since llaFll = Ilv(t) 11 = 1. Therefore the quantity <=,=.) is indepen- will be 2 r. If w([O,ll) does not coincide with 7([O,ll), then we dent of r. But for r = 0 we have easily obtain a strict inequality. f(o,t) = exp = q This completes the proof of 10.4. 9 (0) af ar ar hence =(o,t) = 0. Therefore <=,=> is identically zero, which com- An important consequence of Theorem 10.4 is the following. pletes the proof. COROLL'WY 10.7. Suppose that a path 0: [O,ll -. M, para- Now consider any piecewise smooth curve metrized by arc-length, has length less than or equal to the length of any other path from ~(0)to m( l). Then m m: [a,bl * Uq - (q) . is a geodesic. Each point w(t) can be expressed uniquely in the form exp,(r(t)v(t)) with PROOF: Consider any segment of w lying within an open set W, as 0 < r(t) < E, and Ilv(t)II = 1, v(t) 6 TMq. above, and having length < E.. This segment must be a geodesic by Theorem 10.4. Hence the entire path m is a geodesic. LEMMA 10.6. The length s", 11$11 dt is @eater than Or DEFINITION. A geodesic y: [a,bl -. M will be called minimal if equal to Ir(b) - r(a)1, where equality holds Only if the function r(t) is monotone, and the function v(t) is constant. 62 11. RIEMANNIAN GEOMETRY § 10. GEOD_ESICS AND COMPLETENESS 63 its length is less than or equal to the length of any other piecewise smooth about p. Since S is compact, there exists a point path joining its endpoints. p0 = expp(6v), IIvII = 1, Theorem 10.4 asserts that any sufficiently small segnent of a on S for which the distance to q is minimized. We will prove that geodesic is minimal. On the other hand a long geodesic may not be minimal. expp(rv) = q. For example we will see shortly thht a peat circle arc on the unit sphere This implies that the geodesic segnent t -, 7(t) = expp(tv), is a geodesic. If such an arc has length peater than I[, it is certainly 0 5 t 5 r, not minimal. is actually a minimal geodesic from p to q. In general, minimal geodesics are not unique. For example two anti- The proof will amount to showing that a point which moves along the podal points on a unit sphere are joined by infinitely many minimal geodesics. geodesic 7 must get closer and closer to q. In fact for each t E [6,rl However, the following assertion is true. we will prove that

Define the distance p(p,q) between two points p,q E M to be the (It) ~(7(t),9) = r-t . greatest lower bound for the arc-lengths of piecewise smooth paths joining This identity, for t = r, will complete the proof. these points. This clearly makes M into a metric space. It follows First we will show that the equality (18) is true. Since every easily from 10.4 that this metric is compatible with the usual topologg of M. path from p to q must pass through S, we have

COROWY 10.8. Given a compact set K C M there exists p(p,q) = fin (P(P,s) + p(s,q)) = 6 + p(po,q) . a number 6 > 0 so that any two points of K with dis- SES tance less than 6 are joined by a unique geodesic of Therefore p(p0,q) = r - 6. Since po = 7(6), this prqves (I6). length less than 6. Furthermore this geodesic is minimal; Let to E [6,rl denote the supremum of those numbers t for which and depends differentiably on its endpoints. (It) is true. Then by continuity the equality (1 ) is true also.

PROOF. Cover K by open sets W,, as in 10.3, and let 6 be If to < r we will obtain a contradiction. Let S' denote a small spheri-

small enough so that any two points in K with distance less than 6 lie cal shell of radius 6 about the point 7(t0) ; and let p; E S' be a in a common W,. This completes the proof. point of S' with minimum distance from q. (Compare Diagram lo.) Then Recall that the manifold M is geodesically complete if every geo- P(7(t0),ci) = fin I(~(7(t0),s)+ p(s,q)) = 61 + p(p;,q) , desic segment can be extended indifinitely. S€S hence THEOREM 10.9 (Hopf and Rinow*). If M is geodesically complete, then any two points can be joined by a minimal (2) p(p;,q) = (r - to) - 6' . geodesic. We claim that PA is equal to 7(t0 + sl). In fact the triangle inequality states that PROOF. Given p,q E M with distance r > 0, choose a neighborhood Up as in Lem 10.3. Let C U denote a spherical shell of radius 6 P(P,Pd) 2 P(P,q) - P(P;),q) = to + 6' S P < E (making use of (2)). But a path of length precisely to + 6' * from p to Compare p. 341 of G. de Rham, Sur la reductibilit6 d'un espace de PA obtained bY following 7 fmm P to 7(t0), and then following Helvetici, Vol. 26 (1952); as well as H. Hopf and Riemann,Comentarii Math. a minimal geodesic from Y(t,) to Pd. Since tMs broken geodesic has W. Rinow, Ueber den Begriff der-vollsthdlgen differentialgeometrischen Fliche, minimal length, it follows from Corollary 10.7 that it Is an (unbroken) Cmentarii,Vol. 3 (1931), pp. 209-225. 64 11. FJ3IMANNIAN GEOMETRY § 10. GEODESICS AND COMPLEIENESS 65

and hence coincides with 7. FAMILIAR EXAMPLES OF GEODESICS. In Euclidean n-space, Rn, with Thus 7(t0 + 8') = p;. Now the equality (2) becomes the usual coordinate system x,, ... ,xn and the usual Riemannian metric dxl @ dx, +. . .+ dxn @ dxn we have r& = 0 and the equations for a geo- p(7(t0 + s'),q) = r - (to+ 8') . desic 7, given by t (x, (t),. . . ,xn(t) become and completes the proof. - TM~contradicts the definition of to; 2 T=Oxi J dt whose solutions are the straight lines. TMs could also have been seen as follows: it is easy to show that the formula for arc length

coincides with the usual definition of arc length as the least upper bound

of the lengths of inscribed polygons; from this definition it is clear that straight lines have minimal length, and are therefore geodesics.

The geodesics on Sn are precisely the great circles, that is, the intersections of Sn with the planes through the center of Sn.

PROOF. Reflection through a plane E2 is an isometry I: Sn - Sn Diapam 10 whose fixed point set is C = Sn n E2. Let x and y be two points of C As a consequence one has the following. with a unique geodesic C' of minimal length between them. Then, since I is an isometry, the curve I(C1) is a geodesic of the same length as C' COROLLARY 10.10. If M is geodesically complete then between I(x) = x and I(y) = y. Therefore C' = I(C'). This implies that every bounded subset of M has compact closure. Con- sequently M is complete as a metric space (i.e., every C' c c. Cauchy sequence converges) . Finally, since there is a great circle through any point of Sn in any given direction, these are all the geodesics. PROOF. If X C M has diameter d then for any p E X the map Antipodal points on the sphere have a continium of geodesics of emp: 5-M maps the disk of radius d in TMp onto a compact subset minimal length between them. All other pairs of points have a unique geo- of M which (making use of Theorem 10.9) contains X. Hence the closure desic of minimal length between them, but an infinite family of non-minimal of X is compact. geodesics, depending on how many times the geodesic goes around the sphere Conversely, if M is complete as a metric space, then it is not and in which direction it starts. difficult, using Lemma 10.3, to prove that M is geodesically complete. By the same reasoning every meridian line on a surface of revolution For details the reader is referred to Hopf and Rinow. Henceforth we will is a geodesic. not distinguish between geodesic completeness and metric completeness, but The geodesics on a right circular cylinder Z are the generating will refer simply to a complete Riemannian manifold. lines, the circles cut by planes perpendicular to the generating lines, and 66 11. RIEMANNIAN GEOMETRY 67

the helices on Z.

/---- PART I11

THE CALCULUS OF VARIATIONS APPLIED TO GEODESICS

511. The Path Space of a Smooth Manifold.

Let M be a smooth manifold and let p and q be two (not necessarily distinct) points of M. By a piecewise smooth path from p to q

PROOF: If L is a generating line of Z thenwe can set up an will be meant a map u): [o,ll - M such that 2 isometry I: Z - L *R by rolling Z onto R2: 1') there exists a subdivision 0 = to < t, < ... < tk = 1 of [0,11 so that each 01 [ti-,,ti] is differentiable of class Cm; 2) m(0) = p and ~(l)= q. The set of all piecewise smooth paths from p to q in M will be denoted by R(M;p,q), or briefly by R(M) or R. L I Later (in §16) n will be given the structure of a topological I I ,-- . L space, but for the moment this will not be necessary. We will think of R as being something like an "infinite dimensional manifold." To start the analogy we make the following definition. the straight lines The geodesics on Z are just the images under I-' of By the tangent space of 0 at a path u) will be meant the vector 2 in R . Two points on Z have infinitely many geodesics between them. space consisting of all piecewise smooth vector fields W along u) for which W(0) = 0 and W(1) = 0. The notation TRu) will be used for this vector space.

If F is a real valued function on R it is natural to ask what

F*: Tau) +TRRu)) , the induced map on the tangent space, should mean. When F is a function

which is smooth in the usual sense, on a smooth manifold M, we can define F,: TMp -TRRp) as follows. Given X c TMp choose a smooth path

u -+ a(u) in M, which is defined for -E < u < E , so that ff(0) = P, m(0)dff = x . $1 1 THE PATH SPACE 60 111. CALCULUS OF VARIATIONS . 69 function on il, we attempt to define Then F,(X) is equal to d(Fi:(U))lu=o, multiplied by the basis vector F,: TOw - mq,)

(-&)F(p) mF(P) * as follows. Given W f Taw choose a variation E: (-E,E) + n with In order to carry out an analog9us construction for F: n R, d5 E(0) = a, &O) = w the following concept is needed. and set F,(W) equal to d(F(:(u))u lu=o multiplied by the tangent vector DEFINITION. A variation of o (keeping endpoints fixed) is a . Of course without hypothesis on F there is no guarantee that ( )F(m) function this derivative will exist, or will be independent of the choice of 6: (-E,E) 4 R, 5. We will not investigate what conditions F must satisfy in order for F, for some E > 0, such that to have these properties. 1) 6(0) = o Ere have indicated how F, might be defined only 0 = to tl tk = 1 to motivate the following. 2) there is a subdivision < < ... <

of [o,il SO that the map DEFINITION. A path o is a critical path for a function

a: (-E,E) x [0,11 -+ M F: R - R if and only if luF0 is zero for every variation 6 of is c" on each strip (-E,E) X [ti-,,ti], defined by cr(u,t) = 6(u)(t) o. i = 1, ...,k. EXAMPLE. If F takes on its minimum at a path mo, and if the Since each 6(u) belongs to R = R(M;p,q), note that: derivatives are all defined, then clearly oo is a critical path. 3) O(U,O) = p, a(u,l) = q for all u E (-E,E) .

We will use either a or 5 to refer to the variation. More

generally if, in the above definition, (-E,E.) is replaced by a neighbor-

hood p of 0 in Rn, then a (or 6) is called an n-parameter varia- -tion of a. Now 6 may be considered as a "smooth path" in n. Its "velocity d6 given vector'' =(o) E Trio is defined to be the vector field W along o by an Wt = &It = ,(o,t) *

Clearly W E TRo. This vector field W is also called the variation=- tor field associated with the variation a.

Given any W E TRo note that there exists a variation d5 6: (-E,E) -+ R which satisfies the conditions 5(0) = w, ~(0)= W. In fact one can set

.(u)(t) = expo(t)(u Wt) . By analogy with the definition given above, if F is a real valued 111. OF VARIATIONS $12. THE FNERGY OF A PATH 71 70 CALCULUS takes on its minimum d2 precisely on the set of minimal geodesics from p to q.

$12. The Energy of a Path. We will now see which paths o E 0 are critical paths for the energy function E. Suppose now that M is a Riemannian manifold. The length of a vec- Let E: (-E,E) - be a variation of o, and let Wt = =(o,t)au tor v E TMP will be denoted by \\v(\ = < v,v >*. For w E 0 define the be the associated variation vector field. Furthermore, let: energy of w from a to b (where 0 5 a < b 5 1) as Vt = dw = velocity vector of w ,

At = D dm = acceleration vector of w ,

1 We will write E for Eo. AtV = Vt+ - Vt- = discontinuity in the velocity vector at t, This can be compared with the arc-length from a to b given by where o < t < 1 .

Of course AtV = 0 for all but a finite number of values of t.

a THEOREM 12.2 (First variation formula). The derivative as follows. Applying Schwarz's inequality

a a a PROOF: According to Lemma 8.3, we have with f(t) = 1 and g(t) = 11~11 we see that a aa aCY D aa aa <=,x> = 2 . Theref ore

where equality holds if and only if g is constant; that is if and only if 1 1 d D ~CY~CY the parameter t is proportional to arc-length. du 0 0 Now suppose that there exists a minimal geodesic 7 from p = ~(0) By Lemma 8.5 we can substitute D ~CY in this last formula. to q = ~(1). Then & for Jii Choose 0 = to < tl <...< tk = 1 so that CY is differentiable on E(Y) = L(7)2 5 L(o)~5 E(w) . each strip (-E,E) x [ti-1,ti]. Then we can "integrate by parts" on Here the equality L( 7) = L(w) can hold only if w is also a minimal [ti-l,til, as follows. The identity geodesic, possibly reparametrized. (Compare $10.7.) On the other hand a aa aCY D aCYaa aa D aa the equality L(w) = E( w) can hold only if the parameter is proportional 73€ c3im > = + (2ii.x x > toarc-lengthalong w. This proves that E(7) < E(w) unless w is also implies that a minimal geodesic. In other words:

LFNM!l 12.1. Let M be a complete Riemannian manifold and let p,q E M have distance d. Then the energg function E: n(M;p,q) - R §12. THE ENERGY OF A PATH 73 72 111. CAlLXLUS OF VARIATIONS

PROOF: Clearly a geodesic is a critical point. Let 0) be a Adding up the corresponding formulas for i = 1, ...,k; and using the fact aa critical point. There is a variation of (o with W( t) = f(t)A( t) where that = 0 for t = 0 or 1, this gives f(t) is positive except that it vanishes at the ti. Then 1 1 1 dE(6(u)) = aa D aa 1m - = f(t) A(t),A(t) dt. 2 du ~(0) -! < > i=l 0 0 This is zero if and only if A(t) = c for all t. Hence each (ol[ti,ti+ll Setting u = 0, we now obtain the required formula is a geodesic. 1 Now pick a variation such that W(ti) = at2V. Then

t 0

This completes the proof. a, is differentiable of class C', even at the points ti. Now it follows dE.6 Intuitively, the first term in the expression POP T(o) shows from the uniqueness theorem for differential equations that (o is Cm

everywhere: thus (o is an unbroken geodesic. that varying the path (o in the direction of decreasing "kink," tends to decrease E; see Diagram 11 .

path

// \\

path E(E) I k' smaller em Diagram 11.

The second term shows that varying the curve in the direction cf its D acceleration vector (2) tends to reduce E.

Recall that the path (o E n is called a geodesic if and only if D dm a, is Cm on the whole interval [O, 11, and the acceleration vector m(x)

of a, is identically zero along (o.

COROLLARY 12.3. The path (o is a critical point for the

function E if and only if 0) is a geodesic. S13. THE HESSIAN OF E 74 111. CALCULUS OF VARIATIONS 75

THEOREM 13.1 (Second variation formula). Let 6: U-r n be a 2-parameter variation of the geodesic y with variation vector fields The Hessian of the Energy Function Critical Path. $1 3. at a Wi = xi(O,O)a6 E TRY, i = 1,2 .

Continuing with the analogy developed in the preceding section, we 1 a2E Then the second derivative (0,O) of the energy now wish to define a bilinear functional function is equal to E**: Tn x TO - R Y Y when y is a critical point of the function E, i.e., a geodesic. This

bilinear functional will be called the Hessian of E at y. If f is a real valued function on a manifold M with critical where V = $& denotes the velocity vector field and where point p, then the Hessian DW, DWl Dwl At = E(t+)- x(t-) f,,: T% x TiV$ -r R

can be defined as follows. Given X,,X, c T% choose a smooth map denotes the jump in DW1 at one of its finitely many 2 points of discontinuity in the open unit interval. (u1,u2) + a(ul,u2) defined on a neighborhood of (0,O) in R , with values in M, so that PROOF: According to 12.2 we have

Then Theref ore

This suggests defining E,, as follows. Given vector fields W1,W2 E TnY choose a 2-parameter variation -f< 0 a: U x [O,ll -r M , Let us evaluate this expression where U is a neighborhood of (0,o) in R2, so that an unbroken geodesic, ne have aa 4O,O,t) = Y(t), (O,O,t) = Wl(t), aa (O,O,t) = W2(t) . x1 =2 (Compare 511.) Then the Hessian E*+(W1,W2) will be defined to be the so that the first and third terms are zero. second partial derivative Rearranging the second term, we obtain a2E(E(ul,u2) ) au, au2 1;(0,O) where E(u1,u2) E n denotes the path 6(ul,u2)(t) = a(u1,U2,t) . This a2E In order to interchange the two operators and $ , we need to second derivative will be written briefly as -(O,O) . dU1 U1 u2 bring in the curvature formula, The following theorem is needed to prove that E,, 1s Well defined. 111. CALCULUS OF VARIATIONS 76 $14. JACOB1 FIELDS 77 D D aa D Together with the identity =,V = J--F~, = zwl, this yields

(13.3) 514. Jacobi Fields: The Null Space of E,, Substituting this expression into (1 3.2) this completes the proof of 13.1 . A vector field J along a geodesic y is called a Jacobi field a ‘E COROILARY 13.4. The expression E,,(W1,W2) = .-(o,o) if it satisfies the Jacobi differential equation is a well defined symmetric and bilinear function of W1 and W,. D2J + R(V,J)V = 0 a2E PROOF: The second variation formula shows that -(O,O) dtP where V = . This is a linear, second order differential equation. depends only on the variation vector fields W1 and W,, so that 2 [It can be put in a more familiar form by choosing orthonormal parallel vec- E,,(W1,W2) is well defined. This formula also shows that E,, is bilinear. tor fields Pl, ...,Pn along y. Then setting J(t) = c fi(t)Pi(t), the The symmetry property equation becomes E*,(W1 ,W2) = E**(W2, Wl )

is not at all obvious from the second variation fornula; but does follow a2E - a2E immediately from the symmetry property dU1dU2 - 7q-q . where a? = R(V,P.)V,Pi> .I J < J Thus the Jacobi equation has 2n linearly REMARK 13.5. The diagonal terms E,,(W,W) of the bilinear pairing independent solutions, each of which can be defined throughout 7. The solutions are all (?-differentiable. A given Jacobi field J is com- E,, can be described in terms of a 1-parameter variation of 7. In fact pletely determined by its initial conditions: DJ J(O), ~(0)E TMr(o) *

where 6: (-E,E) + n denotes any variation of 7 with variation vector Let p = y(a) and q = y(b) be two points on the geodesic 7, d5 field =(o) equal to W. To prove this identity it is only necessary to with a f b. introduce the two parameter variation DEFINITION. p and q are conjugate, along y if there exists a B(u1,u2) = 6(Ul + u,) non-zero Jacobi field J along y which vanishes for t = a and t = b. and to note that The multiplicity of p and q as conjugate points is equal to the dimension of the vector space consisting of all such Jacobi fields. As an application of this remark, we have the following. Now let y be a geodesic in R = R(M;p,q). Recall that the s- LEMMA 13.6. If 7 is a minimal geodesic from p to q space of the Hessian then the bilinear pairing E,, is positive semi-definite. E,,: Tny x TO -R Hence the index h of E,, is zero. 7

is the vector space consisting of those W1 E Tn such that E,,(W1,W2) = 0 Y PROOF: The inequality E(G(u)) 2 E(7) = E(E(o)) implies that * If 7 has self-intersections then this definition becomes ambiguous.

One should rather say that the parameter values a and b are conjugate with respect to y. 78 111. CALCULUS OF VARIATIONS 5 14. JACOB1 FIELDS 79

for all W,. The nullity v of E,, is equal to the dimension of this entire unit interval. This completes the proof of 14.1.

null space. E,, is degenerate if v > 0. It follows that the nullity v of E,, is always finite. For

there are only finitely many linearly independent Jacobi fields along 7. THEORFM 14.1. A vector field W1 E Tn belongs to the 7 null space of E,, if and only if W1 is a Jacobi field. REMARK 14.2. Actually the nullity v satisfies 0 5 v < n. Since Hence E,, is degenerate if and only if the end points the space of Jacobi fields which vanish for t = 0 has dimension p and q are conjugate along 7. The nullity of E,, is equal to the multiplicity of p and q as conjugate points. precisely n, it is clear that v 5 n. We will construct one example of a Jacobi field which vanishes for t = 0, but not for

PROOF: (Compare the proof of 12.3.) If J is a Jacobi field which t = 1. This will imply that v < n. In fact let Jt = tVt where vanishes at p and q, then J certainly belongs to The second V = denotes the velocity vector field. Then Tn7 .

($13.1) =J variation formula states that aT-- 14J+tg2 v 1 DV - 7E,,(J,W2) = 1 + f dt = 0 . (Since iTt; = o), hence D2J = 0. Furthermore R(V,J)V = tR(V,V)V t 0 z = 0 since R is skew symmetric in the first two variables. Thus Hence J belongs to the null space. J satisfies the Jacobi equation. Since Jo = 0, J, # 0, this Conversely, suppose that W1 belongs to the null space of E,,. completes the proof. Choose a subdivision 0 = to < tl <...< tk = 1 of [o,ll so that WII [ti-,,til is smooth for each i. Let f: [0,11 - [O,ll be a smooth EXAMPLE 1. Suppose that M is "flat" in the sense that the curva- function which vanishes for the parameter values tO,t,,..., tk and is ture tensor is identically zero. Then the Jacobi equation becomes Dositive otherwise; and let D~J= 0. Setting J(t) = C fi(t)P (t) where Pi are parallel, dt i d2fi this becomes = 0. Evidently a Jacobi field along 7 can have -dt at most one zero. Thus there are no conjugate points, and E,, is Then non-degenerate. - 1 EXAMF'LF 2. Suppose that p and q are antipodal points on the

unit sphere Sn, and let y be a peat circle arc from p to q. Since this is zero, it follows that WII[ti-,,til is a Jacobi field for Then we will see that p and q are conjugate with multiplicity each i. v I DW1 n-1. Thus in this example the nullity of E,, takes its Now let W2 E Tn be a field such that W;(ti) 7 = Ati for largest possible value. The proof will depend on the following = 1,2 ,..., k-1. Then discussion.

Let a be a 1-parameter variation of 7, not necessarily keeping the endpoints fixed, such that each z(u) is a geodesic. That is, let DW1 Hence has no jumps. But a solution W1 of the Jacobi equation is a: (-E,E) x [O,lI +M completely determined by the vectors W1 (ti) and Thus it fol- z(ti). be a Cm map such that a(0,t) = y(t), and such that each 6(u) [given lows that the k Jacobi fields WII [ti-l,til, i = 1, ...,k, fit together by O(u) (t) = a(u,t)l is a geodesic. to give a Jacobi field W1 which is C"-differentiable throughout the 80 111. CALCULUS OF VARIATIONS $14. JACOB1 FIEU)S 81

LEMMA 14.3. If a is such a variation of y through LFXVA 14.4. Every Jacobi field along a geodesic y: [O,ll -M geodesics, then the variation vector field W( t) = %( 0, t) may be obtained by a variation of y through geodesics. is a Jacobi field along y. PROOF: Choose a neighborhood U of y(0) so that any two points D aa PROOF: If a is a variation of 7 through geodesics, then xx of U are joined by a unique minimal geodesic which depends differentiably is identically zero. Hence on the endpoints. Suppose that y(t) E U for 0 5 t 5 6. We will first D D aa D D aa construct a Jacobi field W along yl [o,sl with arbitrarily prescribed

values at t = 0 and t = 6. Choose a curve a: (-E,E) -U so that a(o) = ~(0)and so that g(0) is any prescribed vector in TMr(o). db (Compare 913.3.) Therefore the variation vector field is a Jacobi Similarly choose b: (-E,E) - U with b(0) = y(6) and ,(O) arbitrary. field. Now define the variation Thus one way of obtaining Jacobi fields is to move geodesics around. a: (-€,El [0,61 - M by letting a(u): [O,61 - M be the unique minimal geodesic from a(u)

to b(u). Then the formula t + %(o,t) defines a Jacobi field with the given end conditions.

Any Jacobi field along y([O,sl can be obtained in this way: If

,y(y) denotes the vector space of all Jacobi fields W along 7, then the formula W - (W(O), w(6)) defines a linear map P(Y) +TMy(o) x TMy(6) Now let us return to the example of two antipodal points on a unit n-sphere. Rotating the sphere, keeping p and g fixed, the variation We have just shown that P is onto. Since both vector spaces have the same vector field along the geodesic y will be a Jacobi field vanishing at p dimension 2n it follows that P is an isomorphism. I.e., a Jacobi field and q. Rotating in n-1 different directions one obtains n-1 linearly is determined by its values at y(0) and ~(6).(More generally a Jacobi field is determined by its values at any two non-conjugate points.) There- fore the above construction yields all possible Jacobi fields along

71 [O,6l.

The restriction of Xu) to the interval [0,6l is not essential. If u is sufficiently small then, using the compactness of [o,ll, $u)

can be extended to a geodesic defined over the entire unit interval [0,11. This yields a variation through geodesics:

a': (-€',El) X [O,ll -* M

with any given Jacobi field as variation vector. independent Jacobi fields. Thus p and q are conjugate along 7 with multiplicity n-1 . RFJURK 14.5. This argument shows that in any such neighborhood U the Jacobi fields along a geodesic Segment in U are uniquely determined

1 a2 111. CALCULUS OF VARIATIONS 51 5. THE INDEX THEOREM a3 by their values at the endpoints of the geodesic.

REMARK 14.6. The proof shows also, that there is a neighborhood 515. The Index Theorem. (-6,b) of o so that if t c (-6,6) then 7(t) is not conjugate to The index X of the Hessian 7(o) along 7. We will see in 515.2 that the set of conjugate points to TO7 x TO 7 +R 7(o) along the entire geodesic 7 has no cluster points. E,,: is defined to be the maximum dimension of a subspace of on which E,, TO Y is negative definite. We will prove the following.

THEOREM 15.1 (Morse). The index A of E,, is equal to the number of points 7(t), with 0 < t < 1, such that 7(t) is conjugate to y(0) along 7; each such conjugate point being counted with its multiplicity. This index A is always finite*.

As an immediate consequence one has:

COROWY 15.2. A geodesic segment 7: [0,1 I - M can contain only finitely many points which are conjugate to 7(0) along 7.

In order to prove 15.1 we will first make an estimate for A by splitting the vector space TQ7 into two mutually orthogonal subspaces, on one of which E,, is positive definite.

Each point 7(t) is contained in an open set U such that any two points of U are joined by a unique minimal geodesic which depends differ-

entiably on the endpoints. (See 510.) Choose a subdivision

0 = to < tl <...< tk = 1 of the unit interval which is sufficiently fine so that each sepent y[ti-,,til lies within such an open set U; and so

that each 71 [ti-l,ti3 is minimal. Let Tn7(t0,tl,t2, ...,tk) C TQ7 be the vector space consisting of all vector fields W along 7 such that

1) W\[ti-,,til is a Jacobi field along 71 [ti-l,til for each i;

2) W vanishes at the endpoints t = 0, t = I.

Thus Tn7(to,tl, ..., tk) 1s a finite dimensional vector space consisting of broken Jacobi fields along 7. * For generalization of this result see: W. Ambrose, The index theorem in Riemannian geometrp, Annals Of Mathematics, Vol. 73 (1961), pp. 49-86. 84 111. CALCULUS OF VARIATIONS 5 15. THE INDEX THEOREM

2 Let TI C TR be the vector space consisting of all vector fields 0 5 E,,(W + c W2, W + c Wp) = 2c E,,(W2,W) + c Exx(W2,W2) Y W E TRY which W(to) = 0, W(t,) = 0, W(t2) = 0,. ., W(tk) = 0. for . for all values of c implies that E,,(W,,W) = 0. Thus W lies in the LF,MMA 15.3. The vector space Tny splits as the direct null space. But the null space of E,, consists of Jacobi fields. Since

@ TI. sum Tny(to,tl,. ..,tk) THese two subspaces are TI contains no Jacobi fields other than zero, this implies that W = 0. mutually perpendicular with respect to the inner product Thus the quadratic E,, is positive definite on TI. This E,,. Furthermore, E,, restricted to TI is positive form definite. completes the proof of 15.3. An immediate consequence is the following: PROOF: Given any vector field W E TR let W1 denote the unique Y "broken Jacobi field" in TR (to,tl,.. . ,tk) such that W1 (ti) = W( ti) for Y LEMMA 15.4. The index (or the nullity) of E,, is equal = O,l, ...,k. It follows from 914.5 that W1 exists and is unique. i to the index (or nullity) of E,, restricted to the space Clearly W - W, belongs to TI. Thus the two subspaces, TQy(tO,tl,..., tk) TRy(tO,t,, ..., tk) of broken Jacobi fields. In particular and TI generate TRY, and have only the zero vector field in common. (since TnY(t0,tl, ...,tk) is a finite dimensional vector space) the index A is always finite. If W1 belongs to TRy(to tl,.. .,tk) and W, belongs to TI, then the second variation formula 13.1) takes the form The proof is straightforward. Let yT denote the restriction of y to the interval [O,T~. $E,,(W1,W2) = - 1

second derivative ~ d2E O ' (0); where 6: (-E,E) - R is any variation of du2 ASSERTION (1). X(T) is a monotone function of T, y with variation vector field g(0) equal to W. (Compare 13.5.) If For if T < TI then there exists a X(T) dimensional space ?' of W belongs to TI then we may assume that 15 is chosen so as to leave the vector fields along yT which vanish at y(0) and y(~)such that the Points y(tO),y(tl), ...,y(tk) fixed. In other words we may assume that Hessian ( E: ),, is negative definite on this vector space. Each vector c(u)(ti) = y(ti) for i = 0,l ,..., k. field in 9Y extends to a vector field along yTl which vanishes identically Proof that E,,(W,W) 2 0 for W E TI. Each 6(u) E R is a piece- between y(T) and 7(~').Thus we obtain a A( T) dimensional vector space wise smooth path from ~(0)to y(tl) to y(t2) to ... to ~(1). But Of fields along yT, on which ( E:' ),, is negative definite. Hence each yl[ti-l,til is a minimal geodesic, and therefore has smaller energy X(T) 5 A(T~). than any other path between its endpoints. This proves that ASSERTION (2). A(T) = o for small values of T. E(&(u)) 2 E(Y) = E(C(0)) . For if T is sufficiently small then yT is a minimal geodesic, Therefore the second derivative, evaluated at u = 0, must be 0. 2 hence A(T) = 0 by Lemma 13.6. Proof that E,,(W,W) > 0 for W E TI, W # 0. Suppose that Now let us examine the discontinuities of the function X(T). First E,,(W,W) were equal to 0. Then W would lie in the null space of E,,. note that X(T) 1s Continuous from the left: In fact for any W1 E TR (to,tl,..., tk) we have already seen that ASSERTION (3). For all sufficiently small E > o we have Ex,(Wl,W) = 0. For any W2 E TI the inequality A(T-E) = A(T). 86 111. CALCULUS OF VARIATIONS S15. THE INDEX THEOREM a7

PROOF. According to 15.3 the number X(1) can be interpreted as is negative definite. Let J1,..., Jv be v linearly independent Jacobi the index of a quadratic form on a finite dimensional vector space fields along y,, also vanishing at the endpoints. Note that the v vectors ~n,(t,,t,, ...,tk). We may assume that the subdivision is chosen SO that say ti < T < ti+,. Then the index X(T) can be interpreted as the index of a corresponding quadratic form on a corresponding vector space of H, are linearly independent. Hence it is possible to choose v vector fields broken Jacobi fields along Y,. This vector space is to be constructed X1, ...,Xy along 7T+E so that the matrix using the subdivision 0 < tl < t2 <. . . < ti < T of [O,T]. Since a broken Jacobi field is uniquely determined by its values at the break points

7(ti), this vector space is isomorphic to the direct sum is equal to the v x v identity matrix. Extend the vector fields Wi and

Jh over 7T+E by setting these fields equal to 0 for T 5 t 5 T + E. Using the second variation formula we see easily that

Note that this vector space c is independent of T. Evidently the quadratic form H, on c varies continuously with T. Now H, is negative definite on a subspace '?'C c of dimension ( E:+€)*,( Jh, Xk) = 2~~ (Kronecker delta) A(,). For all TI sufficiently close to T it follows that HTI is Now let c be a small number, and consider the A(T) + v vector fields negative definite on 97. Therefore A(,!) 2 A(,). But if TI = T - E < T then we also have A(,-E) 5 A(,) by Assertion 1. Hence A(,-E) = A(,). w,, ...,WA(,), c-lJ1 - c X, ,..., c-lJV - c Xv

ASSWTION (4). Let v be the nullity of the Hessian ( E: )**. along y,+€. We claim that these vector fields span a vector space of

Then for all sufficiently small E > 0 we have dimension X(T) + v on which the quadratic form ( E:+E),, is negative

X(T+E) = A(?-) + v . definite. In fact the matrix of ( Ei+E),, with respect to this basis is

Thus the function h(t) jumps by v when the variable t passes a conjugate point of multiplicity V; and is continuous otherwise. Clearly this assertion will complete the proof of the index theorem. I c At -4 I + c2 B PROOF that A(,+€) 5 A(,) + v . Let H, and c be as in the proof i of Assertion 3. Since dim ,Z = ni we see that H, is positive definite on where A and B are fixed matrices. If c is sufficiently small, this some subspace 9J1C c of dimension ni - A(,) - V. For all TI sufficient- compound matrix is certainly negative definite. This proves Assertion (4). ly close to T, it follows that HT1 is positive definite on P. Hence The index thearem 15.1 clearly follows from the Assertions (2), (3), and (4).

PROOF that a(,+€) 2 A(,) + V. Let W,,...,WA(,) be X(T) vector fields along y,, vanishing at the endpoints, such that the matrix aa 111. CAEULUS OF VARIATIONS Sl6. AN APPROXIMATION TO Rc 89

PROOF: Let S denote the ball

(X E M : p(x,p) 561 . 816. A Finite Dimensional Approximation to Rc . Note that every path LU E Rc lies within this subset S C M. This follows Let M be a connected Riemannian manifold and let p and q be from the inequality L2 < E 5 c . two (not necessarily distinct) points of M. The set R = n(M;p,q) of Since M is complete, S is a compact set. Hence by 10.8 there piecewise Cm pths p to q can be topologized as follows. Let p from exists E > 0 so that whenever x, y E S and p(x,y) < E there is a

denote the topological metric on M coming from its Riemann metric. Given unique geodesic from x to y of length < E; and so that this geodesic a, mf E 0 with arc-lengths s(t), s'(t) respectively,define the distance depends differentiably on x and y. Choose the subdivision (tO,tl,..., tk) of [o,ll so that each difference ti - ti-l is less than E2/c. Then for each broken geodesic

n(tO,tlj".,tk) C (The last term is added on so that the energy function, we have

will be a continuous function from R to the real numbers.) This metric -< (ti - ti-l)c < E2 . induces the required topology on R. Thus the geodesic cn([ti-l,til is uniquely and differentiably determined by Given c > 0 let Rc denote the closed subset E-'([O,cl) C R 1 the two end points. and let Int nc denote the open subset E-' ( [O,c) ) (where E = EO: R - R The broken geodesic LU is uniquely determined by the (k-l)-tUple is the energy function). We will study the topology of Rc by constructing a finite dimensional approximation to it. u(t,), m(t2),...,cu(tk-l) E M x M x...x M.

Choose some subdivision 0 = toLU(tk-l)) 0: [0,11 -M such that defines a homeomorphism between Int R ( to,tl , ,tk) and a certain open 1) ~(0)= p and (~(1)= q , . . . subset of the (k-l)-fold product M x...x M. Taking over the differentiable 2) m([ti-l,til is a geodesic for each i = 1, .. .,k. Finally we define the subspaces structure from this product, this completes the proof of 16.1. To shorten the notation, let us denote this manifold ci(tO,tl,..., tk)c = nc n R(tO,tl ,..., tk) Int n(to,tl,. . . ,tk)c of broken geodesics by B. Let Int n(to,tl,. ..,tk)c = (Int 0') n n(to, ...,tk) . E': B -R denote the restriction to B of the energy function E : R +R. LEMMA 1.6.1. Let M be a complete Riemannian manifold; and let c be a fixed positive number such that nc # 0. Then for all sufficiently fine subdivisions (tO,tl,..., tk) of [0,1 1 the set Int n(to,tl,. . . ,tk) can be given the structure of a smooth finite dimensional manifold in a natural way. 111. CALCULUS OF VARIATIONS $1 6. AN APPROXIMATION TO Qc 91

THEOREM 16.2. This function El: B -R is smooth. For a < c the set Ba is homeomorphic to the set of all (k-1)- Furthermore, for each a < c the set Ba = (El)-'[o,a] tuples (p, ,..., pkbl) E S x S x.. .x S such that is compact, and is a deformation retract" of the corresponding set Ra. The critical points of E' are precisely the same as the critical points of E in Int nc: namely the unbroken geodesics from p to q (Here it is to be understood that of length less than 6.The index [or the nullity] po = p, pk = 9.) As a closed subset of the Hessian El,, at each such critical point y of a compact set, this is certainly compact. is equal to the index [ or the nullity 1 of E,, at y. A retraction r: Int nc .-L B is defined as follows. Let r(m) Thus the finite dimensional manifold B provides a faithful model denote the unique broken geodesic in B such that each r(m)I[ti-l,til is for the infinite dimensional path space Int Rc. As an immediate conse- a geodesic of length < E from m(ti-l) to .(ti). The inequality p(p,m(t)) 2 5 E < c quence we have the following basic result. (L

implies that m[O,ll C S. Hence the inequality THEOREM 16.3. Let M be a complete Riemannian manifold and let p,q E M be two points which are not conjugate along any geodesic of length 5 &. Then Ra has the homotopy type of a finite CW-complex, with one cell of implies that r(m) can be so defined. dimension X for each geodesic in Ra at which E,, Clearly E(r(m)) 5 E(m) < c. This retraction r fits into a 1- has index X. parameter family of maps (In particular it is asserted that na contains only finitely many ru: Int nc - Int nC geodesics.) as follows. For ti-l 5 u 5 ti let

PROOF. This follows from 16.2 together with 53.5. ru(w) I [O,ti-l I = r(m) I [O,ti-l I ,

16.2. u) E ru(o)I [ ti-l ul = minimal geodesic from "(ti-,) to m(u) , PROOF of Since the broken geodesic B depends smoothly , on the ( k-1 ) -tuple ru(m) I[u,il = ml[u,il . m(tl),o(t2),..., m(tk-l) E M x...~M Then1 ro is the identity map of Int nC, and rl = r. It is easily veri- it is clear that the energy E1(m) also depends smoothly on this (k-1)- fied that ru(m) is continuous as a function of both variables. This proves tuple. In fact we have the explicit formula that B is a deformation retract of Int a'.

Since E(rU(m)) 5 E(m) it is clear that each Ba is also a deformation retract of n". Every geodesic is also a broken geodesic, so it is clear that every 'lcriticai point" of E in Int nc automatically lies in the submanifold B.

Using the i'irst variation formula ($12.2) it is clear that the critical Points of El are precisely the unbroken geodesics.

Consider the tangent space TBy to the manifold B at a geodesic * Similarly B itself is a deformation retract of Int nc. Y. This will be identified with the space TRy(to,tl,...,tk) of broken

96 111. CAXULUS OF VARIATIONS $17. THE FULL PATH SPACE 97 geodesics T~,Y~,Y~,...from p to q, as follows. Let yo denote the REMARK. More generally if M is any complete manifold which 1s short great circle arc p to q; let y1 denote the long great not contractible then any two non-conjugate points of from M are joined by circle arc pq'p'q; let denote the arc pqp'q'pq; and so on. The infinitely many geodesics. r2 Compare p. 484 of J. P. Serre, Homologie singulitSre des espaces fibr6s, Annals of Math. 54 (1951), pp. 425-507.

As another application of 17.4, one can give a proof of the Freuden-

thal suspension theorem. (Compare 522.3.)

subscript k denotes the number of times that p or p' occurs in the interior of rk. The index h(yk) = v1 +...+ pk is equal to k(n-1), since each of the points p or p' in the interior is conjugate to p with multiplicity n-1. Therefore we have:

COROLTARY 17.4. The loop space n(Sn) has the homotopy type of a CW-complex with one cell each in the dimensions 0, n-1, z(n-1), 3(n-1) ,... .

For n > 2 the homology of a(?) can be computed immediately from this information. Since n(Sn) has non-trivial homology in infinitely many dimensions, we can conclude:

COROLLARY 17.5. Let M have the homotopy type of Sn, for n > 2. Then any two non-conjugate points of M are joined by infinitely many geodesics.

This follows since the homotopy type of n*(M) (and hence of n(M)) depends only on the homotopy type of M. There must be at least one geodesic in n(M) with index 0, at least one with index n-1, 2(n-1), 3(n-1), and so on. 918. NON-CONJUGATE POINTS 111. CALCULUS OF VARIATIONS 99 Now suppose that exp, is non-singular at v. Choose n independ-

ent vectors X1, ...,% in T(T )v. Then exp,(X1), ..., exp,(%) linearly independent. In T% Mpchoose paths u - vl(u), ...,u - v are(u) 5.1 8. Existence of Non-Conjugate Points. n with ~~(0)= v and -a+O)dvi (u) = xi . Theorem 17.3 gives a good description of the space n(M;p,q) pro- Then al ,. . .,an, constructed as above, provide n Jacobi fields viding that the points p and q are not conjugate to each other along any yV, W1, ...,Wn along vanishing at p. Since the Wi(l) = exp,(Xi) are This section will justify this result by sharing that such non- geodesic. independent, no non-trivial linear combination of the Wi can vanish at conjugate points always exist. exp v. Since n is the dimension of the space of Jacobi fields along yV, Recall that a smooth map f: N-r M between manifolds of the same which vanish at p, clearly no non-trivial Jacobi field along yv vanishes x E N if the induced map dimension is critical at a point at both p and exp v. This completes the proof.

f,: TNx + TMf(x) COROLLARY 18.2. Let p E M. Then for almost all q c M, 1-1. We will apply this definition to the ex- of tangent spaces is not p is not conjugate to q along any geodesic. ponential map

exp = expp: TMp+M . PROOF. This follows immediately from 18.1 together with Sard's theorem (96.1). (We will assume that M is complete, so that exp is everywhere defined; although this assumption could easily be eliminated.)

THEOREM 18.1. The point exp v is conjugate to p along the geodesic 7,, from p to exp v if and only if the mapping exp is critical at v.

PROOF: Suppose that exp is critical at v E TMp. Then exp,(X)

= 0 for some non-zero X E T(TMp),, the tangent space at V to T, considered as a manifold. Let u- v(u) be a path in T% such that dv v(0) = v and ,(O) = X. Then the map a defined by a(u,t) = exp tv(u)

is a variation through geodesics of the geodesic 7v given by t- exp tv. a is a Jacobi Therefore the vector field W given by t -+ =(exp tv(u)) IuE0 field along yV. Obviously W(0) = 0. We also have

But this field is not identically zero since

So there is a non-trivial Jacobi field along yV from p to exp v,

vanishing at these points; hence p and exp v are conjugate along 7v a 100 111. CALCULUS OF VARIATIONS 61 9. TOPOMGY AND CURVATURE 101

[Intuitively the curvature of a manifold can be described in terms

of "optics" within the manifold as follows. Suppose that we think of the some Relatiops Between TopoloEY and Curvature* 619. geodesics as being the paths of light rays. Consider an observer at p ~u~section will describe the behavior of geodesics in a manifold looking in the direction of the unit vector U towards a point q = exp(rU).

with "negative cumaturelf or with "positive curvature. 'I A small line segment at q with length L, pointed in a direction corre-

sponding to the unit vector W E TS, would appear to the observer as a 19.1. suppose that 0 for line segment of length every pair of vectors A,B in the tangent 3Pce 2

T% and for every p E M. Then no two Points of ~(1+ 5 + (terms involving higher powers of r)) . M are conjugate along any geodesic. Thus if sectional curvatures are negative then any object appears shorter

than it really is. A small sphere of radius E at q would appear to be PROOF. Let 7 be a geodesic with velocity vector field V; and r2 2 an ellipsoid with principal radii E( 1 + TK1 + . . .) , . . ., E (1 + $K, + . . . ) let J be a Jacobi fizld along 7. Then where K1 ,K2,. . . ,K, denote the eigenvalues of the linear transformation 9+ R(V,J)V = 0 W - R(U,W)U. Any small object of volume v would appear to have volume v(1 + T(~lr2 + K, +...+ Q) + (higher terms)) where K~ +...+ K, is equal to the "Ricci curvature" K(U,U), as defined later in this section.] Here are some familiar examples of complete manifolds with curva- Therefore ture 1. 0: J) + 11$112 2 0 - (1) The Euclidean space with curvature 0. (2) The paraboloid z = x2 - y2, with curvature DJ is monotonically increasing, and strictly < 0. Thus the function <= , J > DJ (3) The hyperboloid of rotation x2 + y2 - z2 = 1, with curva- so if # 0. ture < 0. If J vanishes both at 0 and at to > 0, then the function (4) The helicoid x cos z + y sin z = 0, with curvature < 0. < $, J) also vanishes at 0 and to, and hence must vanish identically throughout the interval [O,tOl. This implies that (REMARK. In all of these examples the curvature takes values arbi- J(0) = g(0) = 0, trarily close to 0. It is not known whether or not there exists a complete surface in 3-space with curvature negative and bounded away from so that J is identically zero. This completes the proof. 0.)

A famous example of a manifold with everywhere negative sectional REMARK. If A and B are orthogonal unit vectors at p then the curvature is the pseudo-sphere quantity is called the sectional curvature determined by z = Jl - x2 y2' sech-' A and B. It is equal to the Gaussian curvature of the surface - - + m,z > o with the Riemann metric induced from R3. Here the Gaussian curvature has (ul,u2) * expP (ulA + u2B) the constant value -1. spanned by the geodesics through p with velocity vectors in the subspace No geodesic on this surface has conjugate points although two geo- (see for example, Laugwitz spanned by A and B. "Differential-Geometrie," desics may intersect in more than one point. The pseudo-sphere gives a p. 101.) 81 9. TOPOMGY AND CURVATURE 107 CALCULUS OF VARIATIONS 102 111.

Theref ore, the exponential map expp: T% --L M is one-one and geometry, in which the sum of the angles of any triangle is onto. But it follows from 18.1 that exp is non-critical everywhere; < radians. This manifold is not complete. In fact a theorem of Hilbert P so that exp is locally a diffeomorphism. Combining these two facts, we P see that exp is a global diffeomorphism. This completes the proof of P 19.2. More generally, suppose that M is not simply connected; but is

complete and has sectional curvature 5 0. (For example M might be a flat torus S1 x S1, or a compact surface of genus 2 2 with constant negative curvature.) Then Theorem 19.2 applies to the universal covering space of M. For it is clear that i? inherits a Riemannian metric

from M which is geodesically complete, and has sectional curvature 5 0.

Given two points p,q E M, it follows that each homotopy class of states that no complete surface of constant negative curvature can be paths from p to q contains precisely one geodesic. imbedded in R3. (See e.g. Willmore, "Differential Geometry,'' p. 137.) The fact that 2 is contractible puts strong restrictions on the However, there do exist Riemannian manifolds of constant negative topology of M. For example: curvature which are complete. (See for example Laugwitz, "Differential-

Geometrie," pp. 114-117.) Such a manifold can even be compact; for example, COROLLARY 19.3. If M is complete with (R(A,R)A,B> (Compare Hilbert and Cohn-Vossen, "Anschauliche < 0 then the hornotopy groups ni(M) are zero for a surface of genus 2. 2. - i > 1; and nl(M) contains no element of finite order; Geometrie," p. 228.) other than the identity. THEOREM 19.2 (Cartan*). Suppose that M is a simply COMeCted, complete Riemannian manifold, and that the 1 - PROOF: Clearly ni(M) = ni(M) = 0 for i > 1. Since M is sectional curvature (R(A,B)A,B) is everywhere -< 0. Then any two points of M are joined by a unique geo- contractible the cohomology group Hk (M) can be identified with the co- desic. Furthermore, M is diffeomorphic to the homology group Hk(n,(M)) of the group nl(M). (See for example PP. 200- Euclidean space Rn. 202 of S. T. Hu "Homotopy Theory," Academic Press, 1959.) Now suppose PROOF: Since there are no conjugate points, it follows from the that n,(M) contains a non-trivial finite cyclic subgroup G. Then for a

index theorem that every geodesic from p to q has index A = 0. Thus suitable covering space k of M we have x1 (c) = G; hence Theorem 17.3 asserts that the path space n(M;p,q) has the homotopy type Hk(G) = Hk(k) = 0 for k > n . of a 0-dimensional CW-complex, with one vertex for each geodesic. But the cohomology groups of a finite cyclic group are non-trivial in arbi- The hypothesis that M is simply connected implies that O(M;p,q) trarily high dimensions. This gives a contradiction; and completes the is connected. Since a CoMeCted 0-dimensional CW-complex must consist of proof. a single point, it follows that there is precisely one geodesic from p to Now we will consider manifolds with "positive curvature." Instead 9. * Of considering the sectional curvature, one can obtain sharper results in See E. Cartan, "Lecons sur la GBometrie des Espaces de Riemann," Paris, this case by considering the Ricci tensor (sometines called the "mean curva- 1926 and 1951. ture tensor") . 8 19. TOPOLOGY AND CURVATURE 105 104 111. CALCULUS OF VARIATIONS

DEFINITION. The Ricci tensor at a point p of a Riemannian manifold M is a bilinear pairing 0 UL K: TMp x Ti‘f$ -R 1

defined as follows. Let K(U,,U,) be the trace of the linear transformation S&ng for i = 1, ...,n-1 we obtain W -R(U,,W)U2

from T% to TMp. (In classical terminology the tensor K is obtained from R by contraction.) It follows easily from 99.3 that K is symmetric:

K(U, ,Up) = K(U,,U,) Now if K(Pn,Pn) 2 (n-i)/r2 and L > nr then this expression is Let The Ricci tensor is related to sectional curvature as follows. < 0. Hence E,,(Wi,Wi) < o for some i. This implies that the index of be an orthonormal basis for the tangent space U1,U2, ...,Un T%* 7 is positive, and hence, by the Index Theorem, that y contains conju- ASSERTION. K(Un,Un) is equal to the sum of the sectional curva- gate points. tures < R(Un,Ui)Un,Ui > for i = 1,2,. . . ,n-1 . It follows also that y is not a minimal geodesic. In fact if 5: (-E,E) - n is a variation with variation vector field Wi then PROOF: By definition K(Un,Un) is equal to the trace of the matrix (R(Un,Ui)Un,Uj Since the n-th diagonal term this matrix is dE(C(u)) = o, d2E(E(u)) < , ( > ) . of du du2 zero, we obtain a sum of n-1 sectional curvatures, as asserted. for u = 0. Hence E(n(u)) < E(y) for small values of u # 0. This com- THEOREM 19.4 (Myers*). Suppose that the Ricci curvature pletes the proof. K satisfies K(U,U) 2. (n-l)/r’ EXAMPIX. If M is a sphere of radius r then every sectional for every unit vector U at every point of M; where r curvature is equal to l/r2. Hence K(U,U) takes the constant value is a positive constant. Then every geodesic on M of (n-l)/r2. It follows from 19.4 that every geodesic of length > nr con- length > nr contains conjugate points; and hence is not minimal. tains conjugate points: a best possible result.

PROOF: Let y: [O,ll -c M be a geodesic of length L. Choose COROLLARY 19.5. If M is complete, and K(U,U) 2 parallel vector fields P1, ...,Pn along y which are orthonormal at one (n-l)/r2 > 0 for all unit vectors U, then M is compact, with diameter 5 nr. point, and hence are orthonormal everywhere along y. We may assume that

Pn points along y, so that PROOF. If p,q E M let y be a minimal geodesic from p to 9. DPi V = 3 = LF,, and = 0 . Then the length of y must be 5 nr. Therefore, all points have distance < IT. Since closed bounded sets in a complete manifold are compact, it Let Wi(t) = (sin nt) Pi(t). Then - follows that M itself is compact.

* This corollary applies also to the universal covering space 2 of See S. B. Myers, Riemann manifolds with positive mean Curvatwe, Duke M. Since 2 is COmpaCt, it follows that the fundamental group n,(M) is Math. Journal, Vol. 8 (1941), pp. 401-404. finite. This assertion Can be sharpened as follows. 106 111. CALCULUS OF VARIATIONS $19. TOPOMGY AND CURVATURE 107

THEOREM 19.6. If M is a compact manifold, and if the of positive curvature, auarterly Journal of Mathematics (Oxford), Vol. 7 Ricci tensor K of M is everywhere positive definite, then the path space Q(M;p,q) has the homotopy type of (1936), PP. 316-320. a CW-complex having only finitely many cells in each For other theorems relating topology and curvature, the following dimension. sources are useful.

PROOF. Since the space consisting of all unit vectors U on M Yano and S. Bochner, "Curvature and Betti Numbers," Annals is compact, it follows that the continuous function K(U,U) > 0 takes on K. Studies, No 32, Princeton, 1953. a minimum, which we can denote by (n-l)/r2> 0. Then every geodesic S. S. Chern, On curvature and characteristic classes of a Riemann E n(M;p,q) of length > nr has index X 2 1. manifold, Abh. Math. Sem., Hamburg, Vol. 20 (1955), pp. 117-126. More generally consider a geodesic 7 of length > knr. Then a M. Berger, Sur certaines vari6ti.s Riemanniennes B courbure positive, similar argument shows that 7 has index X 2 k. In fact for each Comptes Rendus Acad. Sci., Paris, Vol. 747 (1958), pp. 1165-1168. i = i,2, ..., k one can construct a vector field Xi along 7 which vanishes S. I. Goldberg, "Curvature and Homology," Academic Press, 1962. outside of the interval ( , ), and such that E,,(Xi,Xi) < 0. Clearly E,,(Xi,Xj) = 0 for i # j; so that XI, ...,Xk span a k- dimensional subspace of TO7 on which E,, is negative definite. Now suppose that the points p and q are not conjugate along any geodesic. Then according to J 16.3 there are only finitely many geodesics from p to q of length 5 knr. Hence there are only.finitely many geodesics with index < k. Together with $17.3, this completes the proof.

REM!UK. I do not know whether or not this theorem remains true if M is allowed to be complete, but non-compact. The present proof certainly 22 breaks down since, on a manifold such as the paraboloid z = x + y , the curvature K(U,U) will not be bounded away from zero.

It would be interesting to know which manifolds can carry a metric so that all sectional curvatures are positive. An instructive example is provided by the product Sm x Sk of two spheres; with m,k 2 2. For this manifold the Ricci tensor is everywhere positive definite. However, the sectional curvatures in certain directions (corresponding to flat tori S' x S' C Sm x Sk) are zero. It is not known whether or not Sm x Sk can be remetrized so that all sectional curvatures are positive. The following partial result is known: If such a new metric exists, then it can not be

invariant under the involution (x,y) --L (-x,-y) of Sm x Sk. This follows

from a theorem of Synge. (See J. L. Synge, On the connectivity Of spaces PART IV

APPLICATIONS TO LIE GROUPS AND SYMMETRIC SPACES

520. Symmetric Spaces.

A symmetric space is a connected Riemannian manifold M such that, for each p E M there is an isometry : M which leaves p fixed IP M - and reverses geodesics through p, i.e., if 7 is a geodesic and 7(O) = p then Ip(7(t)) = 7(-t).

LEMMA k0.1 Let y be a geodesic in M, and let p = y(0) and q = y(c). Then IqIp(y(t)) = 7(t + 2c) (assuming 7(t) and 7(t + 2c) are defined). More- over, IqIp preserves parallel vector fields along 7.

PROOF: Let y'(t) = 7(t + c). Then 7' is a geodesic and

~'(0)= q. Therefore I I (7(t)) = I (y(-t)) = I (7'(-t - c)) = qP 9 q Y'(t + c) = 7(t + 2c).

If the vector field V is parallel along 7 then IP*(V) is

parallel (since Ip is an isometry) and I V(0) = -V(O); therefore P* IppuV(t) = -V( -t). Therefore Iq* I,*(V(t)) = V(t + 2c).

COROLZSlRY 20.2. M is complete. Since 20.1 shows that geodesics can be indefinitely extended.

COROLLARY 20.3. Ip is unique. Since any point is joined to p by a geodesic.

COROLLARY 20.4. If U,V and W are parallel vector fields along 7 then R(U,V)W is also a parallel field along y.

PROOF. If X denotes a fourth parallel vector field along 7, note that the quantity < R(U,V)W,X > is constant along 7. In fact, 110 IV. APPLICATIONS $20. SYMMETRIC SPACES 111

@ven P = 7(0), q = 7(c) , consider the isometry T = 17(c/2)Ipwhich the condition carries p to q. Then R(V,Ui)V = eiUi ,X > < remains true everywhere along 7. Any vector field W along y may be by 20.1. Since T is an isometry, this quantity is equal to expressed uniquely as < R(Up,Vp)Wp,%> . Thus is constant for every parallel W(t) = w,(t)U,(t) +...+ w,(t)Un(t) . vector field X. It clearly follows that R(U,V)W is parallel. Then the Jacobi equation D2w + %(W) = 0 takes the form Manifolds with the property of 20.4 are called locally symmetric. dt2 (A classical theorem, due to Cartan states that a complete, simply connected, d2wi ui + eiwiUi = 0. locally symmetric manifold is actually symmetric.) 1 dt2 1 In any locally symmetric manifold the Jacobi differential equations Since the Ui are everywhere linearly independent this is equivalent to have simple explicit solutions. Let 7: R -* M be a geodesic in a local- the system of n equations ly symmetric manifold. Let V = %(O) be the velocity vector at p = y(0). d2wi

Define a linear transformation 2 + = O * q: T% -+T% We are interested in solutions tht vanish at t = 0. If ei > 0 then

by* %(W) = R(V,W)V. Let el, ...,en denote the eigenvalues of %. wi(t) = ci sin (5t), for some constant ci.

THEOREM 20.5. The conjugate points to p along 7 Then the zeros of wi(t) are at the multiples of t = n/Gi .

are the points y(ck/Gi) where k is any non-zero If ei = 0 then wi(t) = cit and if ei < 0 then integer, and ei is any positive eigenvalue of %. - wi(t) = ci sinh (J leilt) for some constant ci. Thus if ei 1. 0, wi(t) The multiplicity of 7(t) as a conjugate point Is equal to the number of ei such that t is a mul- vanishes only at t = 0. This completes the proof of 20.5. tiple of c/Gi.

PROOF: First observe that % is self-adjoint:

< q(w),w'> = This follows immediately from the symmetry relation

< R(V,W)Vl,Wl) = . Therefore we may choose an orthonormal basis U,, ...,Un for Mp so that

%(Ui) = eiUi ,

where el, ...,en are the eigenvalues. Extend the Ui to vector fields along 7 by parallel translation. Then since M is locally symmetric,

* should not be confused with the Ricci tensor Of $19. §21. LIE GROUPS 113 112 IV. APPLICATIONS

LEMMA 21.2. The geodesics y in G with ~(0)= e are precisely the one-parameter subgroups of G.

521. Lie Groups as Symmetric Spaces. PROOF: Let y: R -+G be a geodesic with 7(O) = e. By Lemma 20.1

the map Iy(t)Ietakes y(u) into 7(u + 2t). Now Iy(t)Ie(~)= y(t)u 7(t) In this section we consider a Lie group G with a Riemannian metric so y(t)~(u)y(t)= y(u + 2t). By induction it follows that y(nt) = y(t)n which is invariant both under left translations for any integer n. If tl/t" is rational so that t' = n't and t" = n"t L,: G + G, L,(u) = T d for some t and some integers n' and n" then y(t' + t") = y(t)n'+n" =

and right translation, R,(u) = UT. If G is commutative such a metric 7(t1)y(t"). By continuity 7 is a homomorphism. certainly exists. If G is compact then such a metric can be constructed Now let y: R - G be a 1-parameter subgroup. Let 7' be the as follows: Let <,> be any Riemannian metric on GI and Let I.I denote geodesic through e such that the tangent vector of 7' at e is the tan-

the Haar measure on G. Then I.I is right and left invariant. Define a gent vector of y at e. We have just seen that 7' is a 1-parameter sub-

new inner product <(,>> on G by group. Hence 7' = 7. This completes the proof. A vector field X on a Lie group G is called left invariant if <> = dp(d) dp(T) . 1 and only if (La)*(Xb) = Xa.b for every a and b in G. If X and Y GxG Then <<,>> is left and right invariant. are left invariant then [X,YI is also. The Lie algebra Q of G is the vector space of all left invariant vector fields, made into an algebra by LEMMA 21.1 If G is a Lie group with a left and right invariant metric, then G is a symmetric space. The the bracket [ I. reflection I, in any point T E G is given by the Q is actually a Lie algebra because the Jacobi Identity formula ~,(u) = To-',. "X,Yl,Zl + "Y,Zl,Xl + "Z,Xl,YI = 0 PROOF: By hypothesis L, and R, are isometries. Define a map holds for all (not necessarily left invariant) vector fields X,Y and Z. Ie: G - G by -1 Ie(a) = u . THEOREM 21.3. Let G be a Lie group with a left and right invariant Riemannian metric. If X,Y,Z and W Then Iex: TGe -+ TGe reverses the tangent space of e; so is certainly are left invariant vector fields on G then: an isometry on this tangent space. Now the identity a) < [x,Yl,Z> = b) R(X,Y)Z = ~[X,Yl,Zl 1, = RU- 1 I,',, - 1 C) = < [X,YI,[Z,W1> . shows that Ie*: Gu - G -1 is an isometry for any u E G. Since 1, reverses the tangent space at e, it reverses geodesics through e. PROOF: As in 50 we will use the notation X k Y for the covariant derivative of Y in the direction X. any left invariant X the iden- Finally, defining IT(u) = TU-~T, the identity I, = R,IeRil For tity shows that each I, is an isometry which reverses geodesics through T. A 1-parameter subgroup of G is a C" homomorphism of R into x FX = 0 G. It is well known that a 1-parameter subgroup of G is determined by is satisfied, since the integral ewes of X are left translates of 1- its tangent vector at e. (Compare Chevalley, "Theory of Lie Groups," Parameter subgroups, and therefore are geodesics. Princeton, 1946.) 114 IV. APPLICATIONS $21 . LIE GROUPS 115

Theref ore COROLLWY 21.4. The sectional curvature = [X,YI, [X,YI is always 0. Equality holds if and (X + Y) I- (X + Y) = (X I- X) + (X l- Y) < > 2 only if [x,YI = 0. + (Y I- X) + (Y I- Y) Recall that the center c of a Lie algebra 0 is defined to be is zero; hence the set of X E 0 such that [X,YI = 0 for all Y E 0. XI-Y+Yl-X = 0.

On the other hand COROLLARY 21.5. If G bs a left and right invariant x kY - Y I- x = [X,YI .. metric, and if the Lie algebra g has trivial center, then G is compact, with finite fundamental group. by 58.5. Adding these two equations we obtain: d) 2X I- Y = [X,YI . PROOF: This follows from Meyer's theorem (819). Let X1 be any

NOW recall the identity unit vector in 0 and extend to a orthonormal basis X1, ...,%. The Ricci curvature Y = + .

(See 58.4.) The left side of this equation is zero, since is K(X1 ,X1) = constant. Substituting formula (d) in this equation we obtain i=1 must be strictly positive, since [Xl,Xil # 0 for some 1. Furthermore 0 = < [Y,XI,Z> + . K(X1 ,X1) is bounded away from zero, since the unit sphere in 0 is compact. Finally, using the skew commutativity of [Y,XI, we obtain the required - * Therefore, by Corollary 19.5, the manifold G is compact. formula This result can be sharpened slightly as follows. <[X,YI,Z> = . COROLLARY 21.6. A simply connected Lie group G with left By definition, R(X,Y)Z is equal to and right invariant metric splits as a Cartesian product - x I- (Y I- 2) + Y I- (X k Z) + [X,YI I- z. G' x Rk where GI is compact and Rk denotes the additive Lie group of some Euclidean space. Furthermore, the Lie SiJbstituting formula (a), this becomes algebra of GI has trivial center. 1 1 1 - F[x,[Y,zll + FIY,[x,zll + T"x,Yl,zl Conversely it is clear that any such product GI x Rk possesses a left and right invariant metric. Using the Jacobi identity, this yields the required formula

R(X,Y)Z = T;"X,YI,ZI1 . (b) PROOF. Let c be the center of the Lie algebra B and let g' = IX E 0 : = o for all C E c 1 The formula (c) follows from (a) and (b) .

be the orthogonal complement of c . Then gr is a Lie sub-algebra. For

if x,y E 01 and C E c then * It follows that the tri-linear function X,Y,Z .-, QX,YI,Z> is skew- symmetric in all three variables. Thus one obtains a left invariant differ- < [X,YI,C > = = 0; ential 3-form on G, representing an element of the de Rham cohomology group hence [x,yI E 0'. It follows that g splits as a direct sum fll (3 C of H3(G). In this way Cartan was able to prove that H3(G) # 0 if G is a non-abelian compact connected Lie group. (See E. Cartan, "La Topologie des Lie algebras. Hence G Splits as a Cartesian product GI x G"; where GI Espaces Representatives des Groupes de Lie," Paris, Hermann, 1936.) is cmpact by 21.5 and. GI' 1s Simply connected and abelian, hence isomorphic $21 LIE GROUPS 116 IV. APPLICATIONS . 117 to some Rk. (See Chevalley, "Theory of Lie Groups.") This completes the by Ad V(W) = [V,WI proof. we have

THEOREM 21.7 (Bott). Let G be a compact, simply con- % = - (Ad V) 0 (Ad V) . nected Lie group. Then the loop space n(G) has the The linear transformation Ad is skew-symmetric; that is homotopy type of a CW-complex with no odd dimensional V cells, with only finitely many X-cells each and for < Ad V(W),W' > = - < W,Ad V(W') > . even value of h. This follows immediately from the identity 21.3a. Therefore we can choose Thus the X-th homology groups of n(G) is zero for A odd, and is an orthonormal basis for G so that the matrix of Ad V takes the form free abelian of finite rank for X even. (-a1O g1-:23* . . REMARK 1. This CW-complex will always be infinite dimensional. As ) an example, if G is the group S3 of unit quaternions, then we have seen that the homology group HiO(S3) is infinite cyclic fgr all even values of i.

REMARK 2. This theorem remains true even for a non-compact group. It follows that the composite linear transformation (Ad V) 0 (Ad V) has In fact any connected Lie group contains a compact subgroup as deformation matrix retract. (See K. Iwasawa, On some types of topological proups, Annals of

Mathematics 50 ( 1949), Theorem 6.)

PROOF of 21.7. Choose two points p and q in G which are not conjugate along any geodesic. By Theorem 17.3, O(G;p,q) has the homotopy Therefore the non-zero eigenvalues of % = - &Ad V)* are positive, and type of a CW-complex with one cell of dimension A for each geodesic from occur in pairs. p to q of index X. By 619.4 there are only finitely many A-cells for It follows from 20.5 that the conjugate points of p along 7 also each X. Thus it only remains to prove that the index X of a geodesic is occur in pairs. In other words every conjugate point has even multiplicity. always even. Together with the Index Theorem, this implies that the index X of any Consider a geodesic 7 starting at p with velocity vector geodesic from p to q is even. This completes the proof. = $(O)ETG 'a. v P According to 620.5 the conjugate points of p on y are determined by the eigenvalues of the linear transformation

TGp -TG q: P' defined by

%(W) = R(V,W)V = $[V,Wl,Vl .

Defining the adjoint homomorphism

Ad V: kl -., $ 22. MANIFOLDS OF MINIMAL GEODESICS 119 118 IV. APPLICATIONS

Thus we obtain: d COROLLARY 22.2. With the same hypotheses, ni(0 ) is isomorphic to IX~+~(M)for 0 i Xo - 2. $22. Whole Manifolds of Minimal Geodesics 5 5

So far we have used a path space n(M;p,q) based on two points Let us apply this corollary to the case of two antipodal points on

However, Bott pointed out Xo = 2n. p,q E K which are in "general position." has the (n+l)-sphere. Evidently the hypotheses are satisfied with that very useful results can be obtained by considering pairs p,q in some For any non-minimal geodesic must wind one and a half times around S""; special position. As an example let M be the unit sphere Sn+l , and and contain two conjugate points, each of multiplicity n, in its interior. let p,q be antipodal points. Then there are infinitely many minimal geo- This proves the following.

In fact the space Rn2 of minimal geodesics forms desics from p to q. COROLLARY 22.3. (The Freudenthal suspension theorem.) a smooth manifold of dimension n which can be identified with the equator The homotopy group ni(Sn) is isomorphic to ni+l(Sn+l) for i 5 2n-2. Sn C Sn+l. We will see that this space of minimal geodesics provides a

P Theorem 22.1 also implies that the homology groups of the loop

space n are isomorphic to those of ad in dimensions 5 Lo - 2. This

fact follows from 22.1 together with the relative Hurewicz theorem. (See for example Hu, p. 206. Compare also J. H. C. Whitehead, Combinatorial homotopy I, Theorem 2.)

The rest of $22 will be devoted to the proof of Theorem 22.1. The proof will be based on the following lema, which asserts that the condition "all critical points have index 2 Xot' remains true when a function is jiggled slightly. 4 Let K be a compact subset of the Euclidean space Rn; let U be fairly good approximation to the entire loop space n(Sn+'). a neighborhood of K; and let

Let M be a complete Riemannian manifold, and let p,c, E M be two f: U-R points with distance p(p,q) = a. be a smooth function such that all critical points of f in K have index -> lo. THEOREM 22.1. If the space ad of minimal geodesics from p to g is a topological manifold, and if every non-minimal LEMMA 22.4. If g: U R is any smooth function which geodesic from p to q has index 2 Lo, then the relative - d is "close" to f, in the sense that homotopy group ni(n,n ) is zero for o 5 i < h0.

It follows that the inclusion homomorphism d ni(O ) ni(n) - uniformly throughout K, for some sufficiently small constant E, is an isomorphism for i 5 Xo - 2. But it is well known that the homotopy then all Critical Points of g in K have index 2 Xo. group ni(n) is isomorphic to ni+l(M) for all values of (Compare i. (Note that f 1s allowed to have degenerate critical points. In S. T. Hu, "Homotopy Theory," Academic Press, 1959, p. 111; together with the application, g will be a nearby function without degenerate critical 817.1.) points.) 120 IV. APPLICATIONS § 22. MANIFOLDS OF MINIMAL GEODESICS 121

PROOF of 22.4. The first derivatives of g are roughly described To complete the proof of 22.4, it is only necessary to show that by the single real valued function the inequalities (*) will be satisfied providing that

for sufficiently small E. This follows by a uniform continuity argument on U; which vanishes precisely at the critical points of g. The second which will be left to the reader derivatives of g can be roughly described by n continuous functions (or by the footnote above ). We will next prove an analogue of Theorem 22.1 for real valued functions on a manifold. as follows. Let Let f: M + R be a smooth real valued function with minimum 0, 1 2 eg(x) 5 eg(x) F... ei(x) such that each MC = f-'[O,cl is compact. a2 denote the n eigenvalues of the matrix Thus a critical point LIDlMA 22.5. If the set Mo of minimal points is a manifold, ( 6-$-xi xj ) . and if every critical point in x of g has index A if and only if the number ek(x) is negative. M - Mo has index 2 Ao, 2 then nr(M,M 0 ) = 0 for 0 5 r < Xo. The continuity of the functions ex follows from the fact that the 63 * A-th eigenvalue of a symmetric matrix depends continuously on the matrix . PROOF: First observe that Mo is a retract of some neighborhood This in turn follows from the fact that the roots of a polynomial depend U C M. In fact Harmer has proved that any manifold Mo is an absolute (See §14 of K. Knopp, "Theory of Functions, continuously on the polynomial. neighborhood retract. (See Theorem 3.3 of 0. Harmer, Some theorems on

Part 11," published in the United States by Dover, 1947.) absolute neighborhood retracts, Arkiv for Matematik, Vol. 1 (1950), pp. LO Let m (x) denote the larger of the two numbers kg (x) and -eg (x). g 389-408.) Replacing U by a smaller neighborhood if necessary, we may Similarly let mf(x) denote the larger of the corresponding numbers kf(x) assume that each point of U is joined to the corresponding point of Mo and -e)(x). The hypothesis that all critical points of f in K have by a unique minimal geodesic. Thus U can be deformed into Mo within M. AO index 2 A. implies that -ef (x) > 0 whenever kf(x) = 0. In other words Let 1' denote the unit cube of dimension r < Xo, and let mf(x) > 0 for all x E K. h: (Ir,Tr) - (M,Mo) Let 6 > 0 denote the minimum of mf on K. Now suppose that g be any map. We must show that h is homotopic to a map h' with is so close to f that h'(Ir) C Mo. (*) Let c be the maximum of f on h(1'). Let 36 > 0 be the mini- all x E K. Then m (x) will be positive x E K; hence every for g for mum of f on the set M - U. (The function f has a minimum on M - U critical point of g in K will have index 2 Lo. since each subset MC - U is compact.) Now choose a smooth function * This statement can be sharpened as follows. Consider two nxn symmetric Mc+26 g: matrices. If corresponding entries of the two matrices differ by at most +R Which approximates f closely, but has no degenerate critical points. This E, then corresponding eigenvalues differ by at most ne. This can be proved using Courant's minimax definition of the A-th eigenvalue. (See Is possible by J6.8. To be more precise the approximation should be so Uber die Ab%n@;igkeit der Schwingungszahlen einer Membr Jl of Courant, an..., close that: Nachrichten, Koniglichen Gesellschaft der Wissenschaften zu Gottingen, Math. (I) If(x) - g(X) 1 < 6 for all x E Mc+26; and Phys. Klasse 1919, pp. 255-264.) 122 IV. APPLICATIONS § 22. MANIFOLDS OF MINIMAL GEODESICS

(2) The index of g at each critical point which lies in the com- Then

pact set f-l[6,C+26] is 2 x0. F 0 E: Int nc(to,t ,,...,tk) + R It follows from Lema 22.4 that any g which approximates f satisfies the hypothesis of 22.5. Hence sufficiently closely, the first and second derivatives also being approxi- ni(Int nc(to ,..., tk),n d E ni(lnt nc,nd) mated, will satisfy (2). In fact the compact set f-’[6,C+261 can be is zero fop i < Lo. This completes the proof. covered by finitely many compact set Ki, each of which lies in a coordinate neighborhood. Lemma 22.4 can then be applied to each %. The proof of 22.5 now proceeds as follows. The function g is smooth on the compact region g-l[26,c+61 c f-’[~,C+~FI, and all critical points are non-degenerate, with index 2 Xo. Hence the manifold g-’ (-m,c+61 has the homotopy type of g-’ (-m,261 with cells of dimension 2 Xo attached. Now consider the map

0 h: Ir,fr -C MC,Mo C g-’(-m,C+6l,M .

Since r < Xo it follows that h is homotopic within g-’ (-m,C+61 ,Mo to a

0 h’: Ir,fr + g-’(-m,26I,M .

But this last pair is contained in (U,Mo); and U can be deformed into Mo within M. It follows that h’ is homotopic within (M,Mo) to a map h”: Ir,fr - Mo,Mo. This completes the proof of 22.5. The original theorem, 22.1, now can be proved as follows. Clearly it is sufficient to prove that

ni(lnt nc,nd) = o

for arbitrarily large values of c. As in $16 the space Int nc contains a smooth manifold Int RC( tO,tl,.. . ,tk) as deformation retract. The space ad of minimal geodesics is contained in this smooth manifold. The energy function E: R +R, when restricted to Int nC(tO,tl, ...,tk), almost satisfies the hypothesis of 22.5. The only difficulty is that E(u) ranges over the interval d E < c, instead of

the required interval [O,m). To correct this, let

F: [d,c) + LO,=) be any diffeomorphism. 124 IV. APPLICATIONS 523. THE BOTT PERIODICITY THEOREM 125

Since U(n) is compact, it possesses a left and right invariant Riemannian metric. Note that the function The Bott Periodicity Theorem for the Unitary Group. 523. exp: TU(nII -U(n) First a review of well known facts concerning the unitary group. defined by exponentiation of matrices coincides with the function exp de- Let Cn be the space of n-tuples of complex numbers, with the usual Her- fined (as in 910) by following geodesics on the resulting Riemannian mani- mitian inner product. The unitary group U(n) is defined to be the group fold. In fact for each skew-Hermitian matrix A the correspondence of all linear transformations S: Cn +Cn which preserve this inner t -L exp(t A) product. Equivalently, using the matrix representation, U(n) is the defines a 1 -parameter subgroup of U(n) (by Assertion (2) above) ; group of all n x n complex matrices S such that S S* = I; where S* and hence defines a geodesic. denotes the conjugate transpose of S.

For any n x n complex matrix A the exponential of A is defined A specific Riemannian metric on U(n) can be defined as follows. Given matrices A,B E g let denote the real part of the complex by the convergent power series expansion 1 number exp A = I + A + &A2 + -A33. + ...

The following properties are easily verified: -1 (1) exp (A*) = (exp A)*; exp (TAT-’) = T(exp A)T . Clearly this inner product is positive definite on g , (2) If A and B commute then This inner product on g determines a unique left invariant exp (A + B) = (exp A) (exp B) . In particular: Riemannian metric on U(n). To verify that the resulting metric is also (3) (exp A)(exp -A) = I right invariant, we must check that it is invariant under the adjoint (4) The function exp maps a neighborhood of 0 in the space of action of U(n) on 0. n x n matrices diffeomorphically onto a neighborhood of I. If A is skew-Hermitian (that is if A + A* = O), then it fol- DEFINITION of the adjoint action. Each S E U(n) determines an inner automorphism lows from (1) and (3) that exp A is unitary. Conversely if exp A is unitary, and A belongs to a sufficiently small neighborhood of 0, then x - s x s-l = (L~R~-’)X = 0. From these facts one it follows from (1) , (3), and (4) that A + A* of the group U(n). The induced linear mapping easily proves that: (5) U(n) is a smooth submanifold of the space of n x n matrices; is called Ad(S). Thus Ad(S) is an automorphism of the Lie algebra of (6) the tangent space TU(n)I can be identified with the space of U(n). n x n skew-Hermitian matrices. Using Assertion (1) above we obtain the explicit formula

Therefore the Lie algebra g of U(n) can also be identified with Ad(S)A = 3M-l ,

the space of skew-Hermitian matrices. For any tangent vector at I extends for A E Q, s E U(n). Computation shows that uniquely to a left invariant vector field on U(n). The inner Product

SO that we may as well assume that A is already in diagonal form implies that

trace (A~B~*)= trace (sAB*s-') = trace (AB") ; and hence that A=

(A1,Bl > = . In this case, is also It follows that the corresponding left invariant metric on U(n) right invariant.

Given A E g we know by ordinary matrix theory that there exists

T E U(n) so that TAT-' is in diagonal form so that exp A = -I if and only if A has the form

there is a T E u(n) where the ails are real. Also, given any S E U(n), for some odd integers k,, ... 7%. Since the length of the geodesic t-exp tA from t = 0 to t = 1

is [A[ = m, the length of the geodesic determined by A is n Jk? +. . .+ c. Thus A determines a minimal geodesic if and only if each $ equals 5 I, and in that case, the length is x &I. Now, regarding that where again the ails are real. Thus we see directly exp: B -+U(n) such an A as a linear map of Cn to Cn observe that A is completely is onto. determined by specifying Eigen(in), the vector space consisting of all in the same way. One may treat the special unitary group SU(n) v c Cn such that Av = inv; and Eigen( -in), the space of all v E Cn

SU(n) is defined as the subgroup of U(n) consisting of matrices of de- such that Av = -inv. Since Cn splits as the orthogonal sum Eigen(in) 6? terminant 1. If exp is regarded as the ordinary exponential map of Eigen(-in), the matrix A is then completely determined by Eigen(in) , matrices, it is easy to show, using the diagonal form, that which is an arbitrary subspace of Cn. Thus the space of all minimal geo-

det (exp A) = etrace A . desics in U(n) from I to -I may be identified with the space of all sub-vector -spaces of Cn. 3sing this equation, one may show that 81 , the Lie algebra of SU(n) is Unfortunately, this space is rather inconvenient to use since it the set of all matrices A such that A + A* = 0 and trace A = 0. and SU(n), has Components of varying dimensions. In order to apply Morse theory to the topology of U(n) This difficulty may be removed by replacing U(n) by SU(n) and setting n = 2m. In this case, all the we begin by considering the set of all geodesics in U(n) from I to -I. above considerations remain valid. But the additional condition that In other words, we look for all A E TU(n)I = g such that exp A = -I. let a, +...+ a - o with ai = 2 n restricts Eigen(in) to being an arbi- Suppose A is such a matrix; if it is not already in diagonal form, 2m - trarg m dimensional sub-vector-space C2m. of This proves the following: T E U(n) be such that TAT-l is in diagonal form. Then

exp TAT-l = T(exp A)T-l = T(-I)T-' = -1 823. THE BOTT PERIODICITY THEOREM 128 IV. APPLICATIONS 129 called the (i-il-st stable homotopy group of the unitary group. LEMMA 23.1. The space of minimal geodesics from I to -I They will 2m) is homeomorphic to the be denoted briefly by in the special unitary group SU( fli-l U. complex Grassmann manifold Gm(Cm), consisting of all m The same exact sequence shows that, for i = 2m+i, the homomorphism dimensional vector subspaces of c2m. nrn U(m) - x2m u(m+i) s n2m U is onto. We will prove the following result at the end of this section. The complex Stiefelmanifold is defined to be the coset space U(m) /V(m). From the exact sequence of the fibration 23.2. Every non-minimal geodesic from I to -I in sU(2m) has index >_ an+2. u(m) -c U(2m) -c U(a)/ u(m)

we see that ni(U(2m)/ u(m)) = o for i < 2m. Combining these two lenunas with 822 we obtain: - The complex Grassmann manifold Gm(Cem) can be identified with THEOREM 23.3 (Bott) The inclusion map Gm(Can) . - the coset space U(2m)/ U(m) x U(m). (Compare Steenrod 87.) From the exact fi(SU(2m); 1,-I) induces isomorphisms of homotopy groups sequence of the fibration in dimensions 5 2m. Hence an ni Gm(C ) r ni+,SU(2m) u(m) - ~(2m)/ U(m) - cm(c2”) for i 5 2111. we see now that Z~G~(C’~)-N- ni-l u(m) On the other hand using standard methods of homotopy theory one

obtains somewhat different isomorphisms. for i 5 2m. Finally, frm the exact sequence of the fibration SU(m) -u(m) - S’ we see that n SU(m) I n U(m) for j + I. his j completes the proof of Lemma 23.4. j Combining Lem 23.4 with Theorem 23.3 we see that

fli-l U = ni-l U(m) r niGm(C 2m ) Y ni+l SU(2m) s “i+l for j # 1. for 1 5 i 5 2m. Thus we obtain: PROOF. First note that for each m there exists a fibration -c S2m+l PERIODICITY THEOREM. U r ni+l U for 12 I. U(m) - U(m+ 1 ) From the homotopy exact sequence To evaluate these groups it is now sufficient to observe that U(i) ... - ni s~~+’- ni-’ u(m) - niV1 U\rn+l) - ni-l Sm+’ - ... I is a circle; SO that no u = no U(1) = 0 of this fibration we see that x1 U = n1 U(1) Z (infinite cyclic). ni-l U(m) s ni-’ U(m+l) for i 5 an. As a check, since SU(2) is a 3-sphere, we have: (Compare Steenrod, “The Topology of Fibre Bundles,” Princeton, 1951, p. 35 n2u = n*SU(2) = 0 and p. 90.) It follows that the inclusion homomorphisms n3 u = n3 SU(2) Y z . u(m+l) ni-, U(m+2) ... we have proved the following result. ni-’ U(m) - - - are all ismorphisms for i 5 an. These mutually iSOmOrPhiC Goups are APPLICATIONS $23. THE BOTT PEiiIODICITY THEOREM 130 IV. 121

THEOREM 23.5 (Bott) . The stable homotopy groups ni u and 2 of the unitary groups are periodic with period 2. In KA(W = ( %(kj - kp)2 wjg) . fact the groups no u r n,U s n4u 2 ... Now we find a basis for g‘ consisting of eigenvectors of KA, as follows: are zero, and the groups For each j < P the matrix Eja with +I in the (jP)-th n 1 u gn3 Urn5Ur ... place, -1 in the (Pj)-th place and zeros elsewhere, is in g’ are infinite cyclic. and is an eigenvector corresponding to the eigenvalue We 2 The rest of §23 will be concerned with the proof of Lemma 23.2. 2 G(k. - kp) J . must compute the index of any non-minimal geodesic from I to -I on Similarly for each j < e the matrix Ejn with +i in the sU(n), where n is even. Recall that the Lie algebra (jQ)-thplace and +i in the (Qj)-thplace is an eigenvector, 2 a’ = T(SU(n))I also with eigenvalue %(kj - ke)2 . Each diagonal matrix in consists of all n x n skew-Hermitian matrices with trace zero. A given 8’ is an eigenvector with eigenvalue 0. Thus the non-zero eigenvalues of KA are the numbers %(2 kj - kQ) matrix A E corresponds to a geodesic from I to -I if and only if

the eigenvalues of A have the form inkl ,..., in% where kl ,... 7% are with k > kp. Each such eigenvalue is to be counted twice. j odd integers with sum zero. Now consider the geodesic r(t) = exp tA. Each eigenvalue 2 I along the geodesic 2 We must find the conjugate points to e = %(kj - ka) > 0 gives rise to a series of conjugate points along y t - exp( tA) . corresponding to the values t = n/&, 2n/v5, 3.1-6, According to Theorem 20.5 these will be determined by the positive eigen- ... . values of the linear transformation (See $20.5.) Substituting in the formula for e, this gives

KA: g’ - g’ t= 4 6 ... . ’j - ’1 ’ kj - ‘P ’ kj - ’P ’ where The number of such values of t in the open interval (0,l) is evidently k -k KA(W) = R(A,W)A = 4 “A,WI,AI . equal to -J-.-& - I. Now let us apply the Index Theorem. For each j,Q with k > kp (Compare $21 .T.) We may assume that A is the diagonal matrix we obtain two copies of the eigenvalue %(kj2 - kQ)2, and hence a jcontri - bution of k. - k 2( +A - 1)

to the index. Adding over all j,Q this gives the formula with k, 2 k2 2 ... 2 %. If W = (w.JP ) then a short computation shows that [A,WI = (in(kj - kQ)wjn) ,

hence for the index of the geodesic 7. [A,[A,WII = (-n2(kj kp)2 Wjp) , - As an example, if y is a IIIinimal geodesic, then all of the kj $24. THE ORTHOGONAL GROW 133 132 IV. APPLICATIONS

are equal to +. 1 Hence A = 0, as was to be expected. . $24. The Periodicity Theorem for the Orthogonal Group. Now consider a non-minimal geodesic. Let n = 2m. This section will carry out an analogous study of the iterated loop CASE 1. At least m+l of the kits are (say) negative. In this space of the orthogonal group. However the treatment is rather sketchy, and case at least one of the positive ki must be 2 3, and we have many details are left out. The point of view in this section was suggested by the paper On Clifford modules (to appear), by R. Bott and A. Shapiro, A2 (3 - (-1) - 2) = 2(m+1) . which relates the periodicity theorem with the structure of certain Clifford 1 algebras. CASE 2. m of the ki are positive and m are negative but not Consider the vector space Rn with the usual inner product. The all are +- 1. Then one is 2 3 and one is -3 so that orthogonal group O(n) consists of all linear maps

A2mf (3 - (-1) - 2) + m$ (1 - (-3) - 2) + (3 - (-3) - 2) T : Rn *Rn 1 1 which preserve this inner product. Alternatively O(n) consists of all

real n x n matrices T such that T T* = I. This group O(n) can be considered as a smooth subgroup of the unitary group U(n); and therefore Thus in either case we have A 2 2m+2. This proves Lemma 23.2, Inherits a right and left invariant Riemannian metric. and therefore completes the proof of the Theorem 23.3. Now suppose that n is even. DEFINITION. A complex structure J on R~ is a linear transfor-

mation J : Rn -Rn, belonging to the orthogonal group, which satisfies

the identity J2 = -I. The space consisting of all such complex structures on Rn will be denoted by n1(n) .

We will see presently (Lemma 24.4) that Ol(n) is a smooth submanifold of the orthogonal group O(n).

REMARK. Given some fixed J, E n,(n) let U(n/2) be the subgroup 9f O(n) consisting of all orthogonal transformations which commute with

J1t Then n,(n) can be identified with the quotient space O(n)mn/e).

LEMMA 24.1. The space of minimal geodesics from I to -I on O(n) is hmemorphic to the space nl(n) of complex structures on R".

PROOF: The space O(n) can be identified with the group of n x n Wthogonai matrices. Its tangent space s = TO(nII can be identified with the space of n x n skew-sgmmetric matrices. Any geodesic 7 with $24. THE ORTHOGONAL GROW 135 134 IV. APPLICATIONS where J* denotes the transpose of J. Together with the identity 7(0) = I can be written uniquely as J J = -I this implies that J* = -J. Thus J is skew-symmetric. Hence

for some A E 9.

Let n = 2m. Since A is skew-symmetric, there exists an element

T E O(n) so that for some &,,a2,... am- > 0 and some T. Now the identity J2 = -I implies that a, = . .. = = 1; and hence that exp nJ = -I. This completes the proof. TAT-’ = -?2*. . ( O:y LEMMA 24.2. Any non-minimal geodesic from I to -I in O(2m) has index 2 2m-2.

-a 0 The proof is similar to that of 23.2. Suppose that the geodesic has

the form t --L exp(nt A) with with a1,a2,. . . ,am2 0. A short computation shows that T(exp n A)T-’ is equal to cos na, sin na, 0 0 ... \ / -sin nal cos “al 0 0 A= *.. \ 0 0 cos na2 sin na2 . ..

0 0 -sin na2 cos 0 are odd integers. Computation shows that the \ ...... non-zero eigenvectors of the linear transformation KA = - & (Ad A)2 are I) for each i < j the number (ai + aj)2/ 4, and

Thus exp(nA) is equal to -I if and only if a1,a2, are odd integers. 2) for each i < j with ai # aj the number (ai - aj)2/ 4. ...,am 22 2 The inner product is easily seen to be 2(a1 + a2 +...+a,). Each of these eigenvalues is to be counted twice. This leads to the formula Therefore the geodesic r(t) = exp(nt A) from I to -I is minimal if and only if a1 = a2 = ... = am = 1. = C (ai + aj - 2) + 1 (ai - aj - 2) . i a If 7 is minimal then j

For a minimal geodesic we have a1 = a2 = ... -&m=’- so that = 0, as expected. For a non-minimal geodesic we have a, 2 3; so that

2 1 (3+1-2) + 0 = 2m - 2. 2 hence A is a complex structure. This completes the proof. Conversely, let J be any complex structure. Since J is orthogo- NOW let us apply Theorem 22.1. The two lennnas above, together with nal we have JJ* = I 136 IV. APPLICATIONS 0 24. THE ORTHOGONAL GROW 137 the statement that a,(n) is a manifold imply the following. PROOF of 24.4. Any point in O(n) close to the identity can be expressed uniquely in the form exp A, where A is a "small," skew- THEOREM 24.3 (Bott) . The inclusion map a,(n) - R O(n) induces isomorphisms of homotopy groups in dimensions sgmmetric matrix. Hence any point in O(n) close to the complex structure -< n-4. Hence J can be expressed uniquely as J exp A,; where again A is small and fii 0,(n) = fli+, O(n) skew. for i .( n-4. ASSERTION 1. J exp A is a complex structure if and only if A Now we will iterate this procedure, studying the space of geodesics anti-commutes with J. from J to -J in nl(n); and so on. Assume that n is divisible by a PROOF: If A anti-commutes with J, then J-lA J = -A hence high power of 2. Let J,, . . . ,Jk-l be fixed complex structures on Rn which anti- I = exp(J-'A J) exp A = J-'(exp A)J exp A . commute *, in the sense that Therefore (J exp A)2 = -I. Conversely if (J exp A)2 = -I then the

JrJs + JsJr = 0 above computation shows that for r # s. Suppose that there exists at least one other complex structure exp(J-lA J) exp A = I . J which anti-commutes with Jl,...,Jk-l. Since A is small, this implies that

1 DEFINITION. Let n,(n) denote the set of all complex structures J J- A J = -A on Rn which anti-commute with the fixed structures . J,,. . . ,Jk-l. so that A anti-commutes with J. Thus we have ASSERTION 2. J exp A anti-commutes with the complex structures n,(n) C C ... C nl(n) C O(n) . J1,...,Jk-l if and only if A commutes with J1,...,Jk-l.

Clearly each n,(n) is a compact set. To complete the definition it is The proof is similar and straightforward. natural to define n0(n) to be O(n) Note that Assertions 1 and 2 both put linear conditions on A. ** Thus a neighborhood of J in %(n) consists of all points J exp A where LEMMA 24.4. Each nk(n) is a smooth, totally geodesic submanifold of O(n). The space of minimal geodesics from A ranges over all small matrices in a linear subspace of the Lie algebra g. Jf to -Jf in O,(n) is homeomorphic to nf+l(n), for This clearly implies that nk(n) is a totally geodesic submanifold of O

It follows that each component of n,(n) is a symetric space. 2 Now choose a specific point Jk E nk(n), and assume that there For the isometric reflection of O(n) in a point of nk(n) will automati- exists a complex structure J which anti-commutes with J1,..., Jk. Setting cally carry nk(n) to itself. J = J@ we see easily that A is also a complex structure which anti- cmutes with Jk. However, A comutes with J1,..., Jk-l. Hence the * These structures make Rn into a module over a suitable Clifford algebra. formula However, the Clifford algebras will be suppressed in the following presen- t -+ Jk exp(nt A) tation. ** defines a geodesic frm Jk to -Jk in llk(n). Since this geodesic is A submanifold of a Riemannian manifold is called totally geodesic if each geodesic in the submanifold is also a geodesic in larger manifold. minimal in o(n), it is certainly minimal in nk(n). 136 IV. APPLICATIONS $24. THE ORTHOGONAL GROUP 139

Conversely, let 7 be any minimal geodesic from Jk to -Jk in Cn/, as being a vector space gl4 over the quaternions H. Let Sp(n/4) a,(n). Setting y(t) = Jk exp(nt A), it follows from 24.1 that A is be the group of isometries of this vector space onto itself. Then n,(n) a complex structure, and from Assertions 1,2 that A commutes with can be identified with the quotient space U(n/2)/ Sp(n/‘c). Jl,... ,Jk-l and anti-commutes with Jk. It follows easily that Jip Before going further it will be convenient to set n = 16r. belongs to Qk+l(n). This completes the proof of 24.4. LEMMA 24.6 - (3). The space a,( 16r) can be identified

REMMRK. The point Jip E flk+l(n) which corresponds to a given with the quaternionic Grassmam manifold consisting of quaternionic subspaces of geodesic y has a very simple interpretation: it is the midpoint y($) of the geodesic. PROOF: Any complex structure J3 en3(16r) determines a splitting In order to pass to a stable situation, note that nk(n) can be of H4r = R’6r into two mutually orthogonal subspaces V, and V, as fol- imbedded in nk(n+nt) as follows. Choose fixed anti-commuting complex lows. Note that JlJ2J3 is an orthogonal transformation with square structures J;,. . . ,Ji on Rn’. Then each J E nk(n) determines a complex J,J2J3J,J2J3 equal to + I. Hence the eigenvalues of J,J,J3 are 1. structure J f3 21; on Rn @Rn’ which anti-commutes with J, @ JA for bt V, C R16r be the subspace on which JlJ2J3 equals + I; and let V,

CY = 1, ...,k-1. be the orthogonal subspace on which it equals -I. Then clearly = V, f3 V,. Since JlJ,J3 commutes with J, and J, it is clear DEFINITION. Let nk denote the direct limit as n-r m of the that both V, and V, are closed under the action of J, and J,. SFaces a,(n), with the direct limit topology. (I.e., the fine topology.) The space 0 =nois called the infinite orthogonal group. . Conversely, given the splitting H4k = V, @ V, into mutually orthogonal quaternionic subspaces, we can define J3 E n3(l6r) by the identities It is not difficult to see that the inclusions -r 0 n,(n) J3P1= -J1J,IV1 give rise, in the limit, to inclusions nk+l -r ank . I J31V2 = JlJ21V2 . THEOREM 24.5. For each k 0 this limit map fik+l 2 - This proves Lemma ?4.6 -(3). 0 ak is a homotopy equivalence. Thus we have isomorphisms (16r) of The space 3 is awkward in that it contains components nh 0 E nhw1 n, z nh-, n, g . . . -= “lnh-l * varying dimension. It is convenient to restrict attention to the component The proof will be given presently. of largest dimension: namely the space of 2r-dimensional quaternionic sub- of Next we will give individual descriptions of the manifolds ak(n) spaces H4r. Henceforth, we will assume that J has been chosen in a 3 for k = 0,1,2, ...,8. this way, so that dimHV1 = dimHV2 = 217. n,(n) is the orthogonal group. LEMMA 24.6 - (4) . The space n4(16r) can be identified nl(n) is the set of all complex structures on Rn . with the set of all quaternionic isometries from Vl to V,. Thus n4(16r) is diffeomorphic to the symplectic Given a fixed complex structure J, we may think of Rn as being a vector group Sp(2~). space Cn/2 over the complex numbers.

n,(n) can be described as the set of “quaternionic structures” on PROOF: Given J,, E n4(16r) note that the product J3J4 anti- the complex vector space Cn’2. Given a fixed J2 en2(n) We my think of Commutes with JlJ,J3 . Hence J3J4 maps V, to V2 (and V, to V1) $24. THE ORTHOGONAL GROUP 141 140 IV. APPLICATIONS splitting of W into two mutually orthogonal subspaces. Let X C W be the Since J3J4 commutes with J, and J, we see that subspace on which J2J4J6 equals +I. Then J,X will be the orthogonal J3J41V, : V, -+v, subspace on which it equals -I. is a quaternionic isomorphism. Conversely, given any such isomorphism Conversely, given X C W, it is not hard to see that J6 is unique- T : V, -V, it is easily seen that J4 is uniquely determined by the ly determined.

identities: REMARK. If O(2r) C U(2r) denotes the poup of complex automor- J41V, = Jil T phisms of W keeping X fixed, then the quotient space U( 2r) / O( 2r) can

J41V2 = -T-’ J3 be identified with a6(16r).

his proves 24.6 - (4) I LFM 24.6 - (7). The space a,( i6r) can be identified LEMMA 24.6 - ( 5) . The space n5(16r) can be identified with the real Grassrnann manifold consisting of real with the set of all vector spaces W C V, such that subspaces of X s Rzr. (I) W is closed under J, (i.e., W is a complex vector space) and (2) V1 splits as the orthogonal sum W 8 J, W. PROOF: Given J7, anti-commuting with J,, ...,J6 note that J,J6J7 commutes with J,J2J3, with J,J4J5, and with J2J4J6; and has

PROOF: Given J5 E a,( 16r) note that the transformation J1J4J5 square +I. Thus J,J6J7 determines a splitting of X into two mutually commutes with J,J,J3 and has square + I. Thus J,J4J5 maps V, into orthogonal subspaces: X, (mere J,J6J7 equals +I) and X, (where

itself; and determines a splitting of V, into two mutually orthogonal sub- J1J6J7 equals -I). Conversely, given X, C X it can be sharn that J7 spaces. Let W C V, be the subspace on which J,J4J5 coincides with + I. Is uniquely determined. Since J2 anti-commutes with J,J4J5, it follows that J2W c V, is This space a7(16r), like n3(16r),has components of varying dimen- precisely the orthogonal subspace, on which J,J4Jg equals -I. Clearly sion. Again we will restrict attention to the component of largest dimen- JIW = W. sion, by assuming that Conversely, given the subspace W, it is not difficult to show that dim X, = dim Xp = r. Thus we obtain: J5 is uniquely determined.

REMARK. If U(2r) C Sp(2r) denotes the group of quaternionic auto- ASSERTION. The largest component of n7(16r) is diffeomorphic to the Grassmann manifold consisting of r-dimensional subspaces of morphisms of V, keeping W fixed, then the quotient swce Sp( 2r) / U( 2r) R2’. can be identified with n5( 16r) . UMNA 24.6 - (8). The space a,( 16r) can be identified with the set of all real isometries from to X,. LEMMA 24.6 - (6). The space a6(161’) can be identified X, with the set of all real subspaces X C W such that W splits as the orthogonal sum X fB J,X. PROOF. If J8 E a8(16r) then the orthogonal transformation J7J8 c-utes with J,J2Jj, J,J4J5, and J2J4J6; but anti-commutes with J,J6J7. PROOF. Given J6 E n6(16r) note that the transformation J,J4J6 &nce JrJ8 maps X, isomorphically onto X,. Clearly this isomorphism commutes both with J,J2J3 and with J1J4Jg. Hence J,J4J6 maps W into determines J8 uniquely.

itself. Since (J,J4J6)2 = I, it follows that J2J4Jg determines a Thus we see that n8(16r) is diffeomorphic to the orthogonal 142 IV. APPLICATIONS $24. THE ORTHOGONAL GROUP 143 group* O(r). DEFINITION. V is a minimal (J,, ...,Jk) -space if no proper, non- Let us consider this diffeomorphism 16r) -+O(r), and pass to a8( trivial subspace is closed under the action of Jl,..., and Jk. Two such the limit as r -c m. It follows that O8 is homeomorphic to the infinite minimal vector spaces are isomorphic if there is an isometry between them Combining this fact with Theorem 24.5, we obtain the orthogonal group 0. which commutes with the action of J,,..., Jk. following. LEMMA 24.8 (Bott and Shapiro) . For k f 3 (mod 4), any 24.7 (Bott) The infinite orthogonal group 0 has THEOREM . two minimal (J1,.,.,Jk) vector spaces are isomorphic. the same homotopy type as its own 8-th loop space. Hence the homotopy group ni0 is isomorphic to ni+8 0 for i 2 0. The proof of 24.8 follows that of 24.6. For k = 0,1, or 2 a minimal space is just a 1-dimensional vector space over the reals, the If Sp =a4denotes the infinite symplectic group, then the above complex numbers or the quaternions. Clearly any two such are isomorphic. argcunent also shows that 0 has the homotopy type of the &-fold loop space nRnR Sp, and that Sp has the homotopy type of the &-fold loop space For k = 3 a minimal space is still a 1-dimensional vector space Rnnn 0. The actual homotopy groups can be tabulated as follows. over the quaternions. However, there are two possibilities, according as J3 is equal to +J1J2 or -J,J2. Thisgivestwo non-isomorphic minimal i modulo ni0 "i 8 1 1 SP spaces, both with dimension equal to 4. Call these H and HI. 0 For k = 4 a minimal space must be isomorphic to H @ HI, with 0 z2 1 0 z2 J3J4 mapping H to HI. The dimension is equal to 8. 2 0 0 For k = 5,6 we obtain the same minimal vector space H @ HI. The 3 Z Z complex structures J5,J6 merely determine preferred complex or real sub- 4 0 Z2 5 0 Z2 spaces. For k = 7 we again obtain the same space, but there are two 6 0 0 possibilities, according as J7 is equal to +J1J2J3J4JgJ6or to 7 Z Z -J1J2J3J4J5J6.Thus in this case there are two non-isomorphic minimal vector spaces; call these L and L'. The verification that these groupsare correct will be left to the reader. For k = 8 a minimal vector space must be isomorphic to L L', (Note that SP(1) is a ?,-sphere, and that SO(3) is a projective 3-space.) with J7J8 mapping L onto L'. The dimension is equal to 16. The remainder of this section will be concerned with the proof of For k > 8 it can be shown that the situation repeats more or less Theorem 24.5. It is first necessary to prove an algebraic lemma. Periodically. However, the cases k 5 8 will suffice for our purposes. Consider a Euclidean vector space V with anti-commuting complex Let mk denote the dimension of a minimal (Jl,..., Jk)-vector space. structures Jl,..., J k' From the above discussion we see that:

mo = 1, ml = 2, m2 = m3 = 4, * For k > 8 it can be shown that ak(l6r) is diffeomorphic to ak-8(r). m4 = mg = m6 = m7 = 8, m8 = 16. In fact any additional complex structures Jg,Jlo,.. . ,Jk on R16r give For k > 8 it can be shown that mk = 16m~-~. rise to anti-commuting complex structures J8Jg, J8Jl0, J8Jll,..., J8Jk on However, for our purposes it REMARK. These numbers mk are closely connected with the problem Xl; and hence to an element of ak-8(r). will be sufficient to stop with k = 8. of constructing linearly independent vector fields on spheres. Suppose for example that J,, ...,Jk are anti-commuting complex structures on a vector F

144 APPLICATIONS IV. § 24. THE ORTHOGONAL GROW 145

space V of dimension rmk. Here r can be any positive integer. Then eigenspaces of A; and hence would not be minimal. Let +_ iah be the two for each unit vector u E V the k vectors Jlu, J2u, ..., Jku are perpen- eigenvalues of AIMh; where al, ...,as are odd, positive integers. dicular to each other and to u. Thus we obtain k linearly independent 1 Now note that Jl = ah JAIMh; is a complex structure on Mh which vector fields on an (rmk-l)-sphere. example we obtain 3 vector For anti-comutes with J1,..., Jk-l, and J. Thus Mh is (J1,..., Jk-,,J,J')- fields on a (4r-l)-sphere; 7 vector fields on an (8r-l)-sphere; vector 8 minimal. Hence the dimension of Mh is mkCl. Since k + 1 f 3 (mod 4) fields on a (16r-I)-sphere; and so on. These results are due to Hurwitz we see that Ml,M2, ...,Ms are mutually isomorphic. and Radon. (Compare B. Echm, Gruppentheoretischer Beweis des Satzes von For each pair h,j with h # j we can construct an eigenvector Hurwitz-Radon ..., Commentarii Math. Helv. Vol. 15 (1g43), pp. 358-366.) J. .' E: Rn -Rn of the linear transformation KA: T - T as follows. Let F. Adam has recently proved these estimates are best possible. that BIMp be zero for P # h,j. Let BIMh be an isometry from Mh to M j which satisfies the conditions PROOF of Theorem 24.5 for k f 2 (mod 4). We must study non- BJ, = JJ! for a = 1, ...,k -1; minimal geodesics from J to -J in nk(n). Recall that the tangent space E?J = -JB and BJ' = +JIB . of nk(n) at J consists of all matrices J A where

1) A is skew In other words B(Mh is an isomorphism from Mh to Mj; where the bar in-

2) A anti-commutes with J dicates that we have changed the sign of J on Mj. Such an isomorphism 3) A commutes with J,, ...,Jk-,. exists by 24.8. Finally let BIMj be the negative adjoint of B(Mh. Proof that belongs to the vector space T. Since Let T denote the vector space of all such matrices A. A given A E T B

corresponds to a geodesic t - J exp (nL4) from J to -J if and only if

KA B = -+ [A,[A,BII = (-A2B + PABA - BA2)/4 , We claim that B is an eigenvector of KA corresponding to the eigenvalue (ah + aj)2/4. For example if v E Mh then just as before. We must construct some non-zero eigenvalues of KA so as (K~B)~= (-A~B+ ~ABA- m2)v to obtain a lower bound for the index of the corresponding geodesic + 12 2 = (a Bv + 2a Ba v + Bah)v t -, J exp(nt A) . j jh = Ti1 (aj + ah)2 BV ;

Split the vector space Rn as a direct sum M, 8 M2 Q . . . Q M, of and a similar computation applies for v E M j* mutually orthogonal subspaces which are closed and minimal under the action Now let us count. The number of minimal spaces Mh C Rn is given of J, Jk-l, J and A. Then the eigenvalues of A on Mh must be ,..., by s = n/mk+l. For at least one of these the integer &n must be 2 3. all equal, except for sip.* For otherwise Mh would split as a sum of For otherwise we would have a minimal geodesic. This proves the following * (always for k f 2 (mod 4) ) : We are dealing with the complex eigenvalues of a real, skew-symmetric transformation. Hence these eigenvalues are pure imagiinars; and occur in ASSERTION- KA at least 9-1 eigenvalues which are conjugate pairs. -> (3+112/4 = 4. he integer s = n/mk+l tends to infinity with n. 146 IV. APPLICATIONS § 24. T.JE ORTHOGONAL GROUP 147

Now consider the geodesic t -+ J exp(nt A). Each eigenvalue e2 NOW consider a geodesic of KA gives rise to conjugate points along this geodesic for t - J exp(ntA) t = e-l , 2e-l, 3e-’, . . . by 20.5. Thus if e2 2 4 then one obtains at from J to -J in nk(n). Since A commutes with i = JIJ, . . . Jk-’ least one interior conjugate point. Applying the index theorem, this proves (compare Assertion 2 in the proof of 24:b) we may think of A also as a the following. complex linear transformation. In fact A is skew-Hermitian; hence the trace of A is a pure imaginary number. Now ASSERTION. The index of a non-minimal geodesic from J to -J in nt trace A nk(n) is 2 n/mk+’- I. f(J exp(ntA)) = determinant (exp(ntA)) = e

It follows that the inclusion map Thus f maps the given geodesic into a closed loop on S’ which is completely determined by the trace of A. It follows that this trace is in-

fik+l(n) -+ n %(n) variant under homotopy of the geodesic within the path space n(nk(n);J,-J) . induces isomorphisms of homotopy groups in dimensions L n/mk+l - 3. This The index X of this aeodesic- can be estimated as follows. As number tends to infinity with n. Therefore, passing to the direct limit before split Rn into an orthogonal sum M, @ . . . G3 where each Mh as n-m, it follows that the inclusion map i : Qk+, -+ n % induces is closed under the action of Jl,...,Jk-l,J, and A; and is minimal. isomorphisms of homotopy groups in all dimensions. But it can be shown Thus for each h, the complex linear transformation AIMh can have only that both nk+’ and 12 % have the homotopy type of a CW-complex. There- one eigenvalue, say iah. For otherwise Mh would split into eigenspaces. fore, by Whitehead’s theorem, it follows that i is a homotopy equivalence. Thus AIMh coincides with ahJ1J2 ... Jk-lIMh. Since Mh is minimal under This completes the proof of 24.5 providing that k $ 2 (mod 4). the action of Jl,...,Jk-l, and J; its complex dimension is mk/2. ~~ PROOF of 24.5 for k 2 (mod 4). The difficulty in this case may Therefore the trace of A is equal to i(al+...+ ar)mk/2. be ascribed to the fact. that n,(n) has an infinite cyclic fundamental Now for each h # j an eigenvector B of the linear transforma- group. Thus n fik(n) has infinitely many components, while the approximat- tion ing subspace nk+,(n) has only finitely many. B -+ KAB = (-A2B + PABA - BA’) /4

To describe the fundamental group n&(n) we construct a map Can be constructed much as before. Since Mh and M. are (J1,...,Jk-’,J)- J f : nk(n) - S’ c c minimal it follows from 24.8 that there exists an isornet-y BIMh : Mh-+ M. as follows. Let J,, ...,Jk-, be the fixed anti-commuting complex struc- J ture on Rn. Make Rn into an (n/2)-dimensional complex vector space by which commutes with Jl,...,Jk-l and anti-commutes with J. Let BIMj be defining the negative adjoint of BIMh; and let BIMp be zero for e # h,j. Then iv = J,J2 ... Jk-iv an easy computation shows that for v E Rn; where i = clE C. The condition k I 2 (mod 4) guarantees KAB = (ah - aj)2B/4 . that i2= -1, and that J,,J, ,..., Jk-l commute with i. Thus for each ah > aj we obtain an eigenvalue (ah - aj)‘/4 for KA. Choose a base point J E fi,(n). any J‘ E nk(n) note that the For Since each such eigenvalue makes a contribution of (ah - a.)/2 - 1 J with i. J-lJ’ composition J-’J’ commutes Thus is a unitary complex towards the index X, we obtain the inequality linear transformation, and has a well defined complex determinant which will 2X (ah - aj - 2) . be denoted by f(J‘). 1. 1 ah> &j APPLICATIONS 148 IV. 149

Now let us restrict attention to some fixed component of nClk(n). That is let us look only at matrices A such that trace A = icmk/2 where c is some constant integer. APPENDIX. THE HOMOTOPY TYPE OF A MONOTONE UNION Thus the integers al, ...ar satisfy

1) a1 = a2 I ... ar E 1 (mod 2), (since exp(nA) = -I), The object of this appendix will be to give an alternative version 2) a1 +...+ a = c, and for the final step in the proof of Theorem 17.3 (the fundamental theorem a 3) Max Iahl 2 3 (for a non-minimal geodesic). of Morse theory). Given the subsets R C nal C na2 C ... of the path h space a = n(M;pJq), and given the information that each nai has the Suppose for example that some ah is equal to -3. Let p be the sum of homotopy type of a certain CW-complex, we wish to prove that the union R the positive ah and -9 the sum of the negative ah. Thus also has the homotopy type of a certain CW-complex. P-9 = c, ~+q>r, More generally consider a topological space X and a sequence hence 2p 2 r + c. Now X, C X1 C X2 C ... of subspaces. To what extent is the homotopy type of X determined by the homotopy types of the Xi? 2~ (ah - aj - 2) > (ah - (-3) - 3) = P , 2 1 1 It is convenient to consider the infinite union %>aj ah > 0 x, = xo XIOJll " xlx [1,21 u x2 x[2,31 u ... . hence 4X 2 2p 2 r + c; where r = n/mk tends to inflnity with n. It follows that the component of n %(n) is approximated up to higher and This is to be topologized as a subset of X x R. higher dimensions by the corresponding component of (n), as n - m . Passing to the direct limit, we obtain a homotopy equivalence on each com- DEFINITION. We will say that X is the homotopy direct limit of

ponent. This completes the proof of 24.5. the sequence [Xi) if the projection map p : X, + X, defined by

p(x,.r) = x, is a homotopy equivalence.

EXAMPLE 1. Suppose that each point of X lies in the interior of some Xi, and that X is paracompact. Then using a partition of unity one can construct a map

f : X+R

SO that f(x) 2 i+l for x q! XiJ and f(x) 2 0 for all x. Now the correspondence x+ (x,f(x)) maps X homeomorphically onto a subset of X, which is clearly a deformation retract. Therefore p is a homotopy equivalence; and X is a homotopy direct limit.

EXAMPLE 2. Let X be a CW-complex, and let the Xi be subcomplexes with union X. Since P : Xz - X induces isomorphisms of homotopy groups in all dimensions, it follows frm Wtehead's theorem that X is a homotopy direct limit. 150 APPENDIX HOMOTOPY CF A MONOTONE UNION 151

EXAMPL3 3. The unit interval [0,11 is the homotopy direct not as follows (where it is always to be understood that 0 5 t 5 1, and limit of the sequence of closed subsets LO1 [l/i,ll. n = 0,1,2 ,...).

The main result of this appendix is the following.

THEOREM A. Suppose that X is the homotopy direct limit of (Xi] and Y is the homotopy direct limit of (Yi). Let f: X+ Y be a map which carries each Xi into Yi by a homotopy equivalence. Then f Taking u = 0 this defines a map : + X, which is clearly homotopic itself is a homotopy equivalence. hO %

to f,. The mapping hl : X, + X, on the other hand has the following

Assuming Theorem A, the alternative proof of Theorem 17.3 can be proper t ies : given as follows. Recall that we had constructed a commutative diagram hl(x,n+t) = (x,n+2t) for o

hl(x,n+t) E X,+l x [n+ll for 3 5 t 5 1 .

KO C K, C K2 C ... We will show that any such map hl is a homotopy equivalence. In fact a homotopy inverse g : X, - $ can be defined by the formula of homotopy equivalences. Since n = U nai and K = U Ki are homotopy direct limits (compare Examples 1 and 2 above), it follows that the limit f (x,n+2t) o

PROOF of Theorem A. Define f, : - Y, by f,(x,t) = (f(x),t). This is well defined since It is clearly sufficient to prove that fz is a homotopy equivalence. hl(x,n++) = hl(x,n+l) = (x,n+l) .

CASE 1. Suppose that Xi = Yi and that each map fi : Xi+ Yi Proof that the composition hlg is homotopic to the identity map (obtained by restricting f) is homotopic to the identity. We must prove Of X,. Note that (x,n+4t) o

This is well defined since ($1, and conclude that Gf, is a homotopy equivalence. This proves that fz has a left homotopy inverse. A similar hlg(x,n+(l-u)/2) = hlg(x,n+$+u) = hl(x,n+l-u). argument shows that fzG : Y, - Yc is a homotopy equivalence, so that Now Ho is equal to hlg and HA is given by f, has a right homotopy inverse. This proves that f, is a homotopy equivalence (compare page 22) and completes the proof of Theorem A.

COROLLAFZ. Suppose that X is the homotopy direct limit of (Xi). If each Xi has the homotopy type Clearly this is homotopic to the identity. of a CW-complex, then X itself has the homotopy Thus hlg is homotopic to the identity; and a completely analogous type of a CW-complex. argument shows that ghl is homotopic to the identity. This completes the The proof is not difficult. proof in Case 1.

CASE 2. Now let X and Y be arbitrary. For each n let $I : Yn- be a homotopy inverse to fn. Note that the diagram

$I 'n xn

] jn $I+1 lin 'n+ 1 ' %+l

(where in and jn denote inclusion maps) is homotopy commutative. In fact

in% * %+lfn+iin% = h+iJnfn& %+iJn .

Choose a specific homotopy with h: = in%, hy = jn; $ : Yn + XnC1 gn,, and define G : Yc -% by the formula (%(Y) ,n+2t) o

We will show that the composition Gf, : 3 + X, is a homotopy equivalence. Let X: denote the subset of X, consisting of all pairs (x,r) with

T 5 n. (Thus $ = Xo x[O,ll u .. . s-lx [n-1 ,nl u Xn x[nl.) The composition Gf, carries into itself by a mapping which is homotopic to the identity. In fact contains Xnx [nl as deformation retract; and the mapping Gf, restricted to Xnx [nl can be identified with %fn, and hence is homotopic to the identity. Thus we can apply Case 1 to the sequence