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MORSE THEORY BY J. Milnor Based on lecture notes by M. SPIVAK and R. WELLS ANNALS OF MATHEMATICS STUDIES PRINCETON UNIVERSITY PRESS Annals of Mathematics Studies Number 51 MORSE THEORY BY J. Milnor Based on lecture notes by M. SPIVAK and R. WELLS PRINCETON, NEW JERSEY PRINCETON UNIVERSITY PRESS Copyright © 1963, © 1969, by Princeton University Press All Rights Reserved L.C. Card 63-13729 ISBN 0-691-08008-9 Third Printing, with corrections and a new Preface, 1969 Fourth Printing, 1970 Fifth Printing, 1973 Printed in the United States of America PREFACE This book gives a present-day account of Marston Morse's theory of the calculus of variations in the large. However, there have been im- portant developments during the past few years which are not mentioned. Let me describe three of these R. Palais and S. Smale nave studied Morse theory for a real-valued function on an infinite dimensional manifold and have given direct proofs of the main theorems, without making any use of finite dimensional approximations. The manifolds in question must be locally diffeomorphic to Hilbert space, and the function must satisfy a weak compactness condition. M As an example, to study paths on a finite dimensional manifold one considers the Hilbert manifold consisting of all absolutely con- tinuous paths w: (0,11 - M with square integrable first derivative. Ac- counts of this work are contained in R. Palais, Morse Theory on Hilbert Manifolds, Topology, Vol. 2 (1963), pp. 299-340; and in S. Smale, Morse Theory and a Non-linear Generalization of the Dirichlet Problem, Annals of Mathematics, Vol. 8o (1964), pp. 382_396. The Bott periodicity theorems were originally inspired by Morse theory (see part IV). However, more elementary proofs, which do not in- volve Morse theory at all, have recently been given. See M. Atiyah and R. Bott, On the Periodicity Theorem for Complex Vector Bundles, Acts, Mathematica, Vol. 112 (1964), pp. 229_247, as well as R. Wood, Banach Algebras and Bott Periodicity, Topology, 4 (1965-66), pp. 371-389. Morse theory has provided the inspiration for exciting developments in differential topology by S. Smale, A. Wallace, and others, including a proof of the generalized Poincare hypothesis in high dimensions. I have tried to describe some of this work in Lectures on the h-cobordism theorem, notes by L. Siebenmann and J. Sondow, Princeton University Press, 1965. Let me take this opportunity to clarify one term which may cause confusion. In §12 I use the word "energy" for the integral v vi PREFACE 1 2 E = s u at II at 0 along a path w(t). V. Arnol'd points out to me that mathematicians for the past 200 years have called E the "action"integral. This discrepancy in terminology is caused by the fact that the integral can be interpreted, in terms of a physical model, in more than one way. Think of a particle P which moves along a surface M during the time interval 0 < t < 1. 'Tie action of the particle during this time interval is defined to be a certain constant times the integral E. If no forces act on P (except for the constraining forces which hold it within M), then the "principle of least action" asserts that E will be minimized within the class of all paths joining w(0) to w(1), or at least that the first variation of E will be zero. Hence P must traverse a geodesic. But a quite different physical model is possible. Think of a rubber band which is stretched between two points of a slippery curved surface. If the band is described parametrically by the equation x = w(t), 0 < t < 1, then the potential energy arising from tension will be proportional to our integral E (at least to a first order of approximation). For an equilibrium position this energy must be minimized, and hence the rubber band will describe a geodesic. The text which follows is identical with that of the first printing except for a few corrections. I am grateful to V. Arnol'd, D. Epstein and W. B. Houston, Jr. for pointing out corrections. J.W.M. Los Angeles, June 1968. CONTENTS PREFACE v NON-DEGENERATE SMOOTH FUNCTIONS ON A MANIFOLD PART I. Introduction . . . . . . . . . . . . . . . §2. Definitions and Lemmas . . . . . . . . . . §3. Homotopy Type in Terms of Critical Values §1. §4. Examples. . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . 4 . . . . . . . . 12 . . . . . . . . 25 . . . . . 28 §5. The Morse Inequalities . . . . . . . . . . . . . §6. Manifolds in Euclidean Space: The Existence of Non-degenerate Functions . §7. PART II. . . . . . . . 32 The Lefschetz Theorem on Hyperplane Sections. . . . . . . 39 . . . . . . . . . A RAPID COURSE IN RIEMANNIAN GEOMETRY §8. Covariant Differentiation . . . . . . . . . . . . . . . . 43 §9. The Curvature Tensor . . . . . . . . . . . . . . . . . 51 §10. Geodesics and Completeness . . . . . . . . . . . . . . . . 55 . . PART III. 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: . 77 §15. The Index Theorem . . 82 §16. A Finite Dimensional Approximation to §17. The Topology of the Full Path Space §18. Existence of Non-conjugate Points §19. Some Relations Between Topology and Curvature . . . . . . . The Null-space of . . . vii . . . . . . E... . . . . . . . . . . . . . . . . . . . . . . . 88 . . . . . . . . 93 . . . . . . . . 98 . . . . sic . . . . . . . . . . . 100 CONTENTS APPLICATIONS TO LIE GROUPS AND SYMMETFIC SPACES PART IV. §20. Symmetric Spaces §21. Lie Groups as Symmetric Spaces §22. Whole Manifolds of Minimal Geodesics §23. The Bott Periodicity Theorem for the Unitary Group §24. The Periodicity Theorem for the Orthogonal Group. APPENDIX. . . . . . . . . . . . . . THE HOMOTOPY TYPE OF A MONOTONE UNION viii . . . . . . . . . . . . log 112 . . . . . . . . . . . . . . . . . . . . 118 . . 124 . . . . . . . . . . . . . . 133 149 PART I NON-DEGENERATE SMOOTH FUNCTIONS ON A MANIFOLD. §1. Introduction. In this section we will illustrate by a specific example the situation that we will investigate later for arbitrary manifolds. sider a torus M, tangent to the plane Let us con- as indicated in Diagram 1. V, Diagram 1. Let f: M -f R (R always denotes the real numbers) be the height above the V plane, and let f(x) < a. Ma be the set of all points x e M such that Then the following things are true: (1) If a < 0 = f(p), (2) If f(p) < a < f(q), then Ma is homeomorphic to a 2-cell. (3)' If f(q) < a < f(r), then Ma is homeomorphic to a cylinder: (4) If Ma then f(r) < a < f(s), then Ma is vacuous. is homeomorphic to a compact manifold of genus one having a circle as boundary: 1 I. 2 (5) If NON-DEGENERATE FUNCTIONS then f(s) < a, Ma' is the full torus. In order to describe the change in Ma of the points f(p),f(q),f(r),f(s) a passes through one it is convenient to consider homotopy type rather than homeomorphism type. (1) as In terms of homotopy types: is the operation of attaching a 0-cell. homotopy type is concerned, the space Ma, For as far as f(p) < a < f(q), cannot be dis- tinguished from a 0-cell: Here means "is of the same homotopy type as." 1-cell: (2) -* (3) is the operation of attaching a (3) - (4) is again the operation of attaching a 1-cell: (4) -. (5) is the operation of attaching a 2-cell. The precise definition of "attaching a k-cell" can be given as follows. Let Y be any topological space, and let ek = {x ERk : 1xII < 1) be the k-cell consisting of all vectors in Euclidean k-space with length < 1. P. INTRODUCTION 3 The boundary ek will be denoted by Sk-1. = If (x E Rk g: Sk-1 IIxII = 1) -+ Y is a continuous map then Y .gek (Y with a k-cell attached by g) is obtained by first taking the topologi- cal sum (= disjoint union) of Y and with x E Sk-1 point and let g(x) E Y. 60 = S-1 and then identifying each ek, To tale care of the case k = 0 let be a eo be vacuous, so that Y with a 0-cell attached is just the union of Y and a disjoint point. As one might expect, the points topy type of Ma' p,q,r At both zero. that x2 - p we can choose f = constant y 2. -x2 - y2, so that (x,y) and at q and f. If we choose any coordinate near these points, then the derivatives (x,y) at which the homo- s changes, have a simple characterization in terms of They are the critical points of the function. system and r (Tx and y are f = x2 + y2, so that at so s f = constant + Note that the number of minus signs in the expression for each point is the dimension of the cell we must attach to go from Ma Mb, where a < f(point) < b. at f to Our first theorems will generalize these facts for any differentiable function on a manifold. REFERENCES 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, "Variationsrechnung its Grossen," published in the United States by Chelsea, New York, 1951. R. Bott, The stable homotopy of the classical groups, Annals of Mathematics, Vol. 70 (1959), pp. 313-337. R. Bott, Morse Theory and its application to homotopy theory, Lecture notes by A. van de Ven (mimeographed), University of Bonn, 1960. 4 NON-DEGENERATE FUNCTIONS I. Definitions and Lemmas. §2. The words "smooth" and "differentiable" will be used interchange- ably to mean differentiable of class manifold M at a point smooth map with p e M (x',...,xn) is zero. of a J this means that _ ... = of (P) axn (p) = 0 . Ma f A critical point f. x e M the set of all points such that p f(x) < a. then it follows from the implicit is a smooth manifold-with-boundary. is a smooth submanifold of f- I(a) p of is not a critical value of function theorem that A if the induced map is called a critical value of f(p) We denote by Ma If f M. If we choose a local coordinate system in a neighborhood U ax is a TNq. is called a critical point of The real number g: M -+ N be a smooth real valued function on a manifold f f*: TMp -T Rf(p) If TMp. then the induced linear map of tangent spaces g,: TMp Now let point will be denoted by p g(p) = q, will be denoted by The tangent space of a smooth C". The boundary M. is called non-degenerate if and only if the matrix a2f (p)) axiaxj 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 intrinsic definition. If functional then v p f** is a critical point on TMp, of f called the Hessian of and w have extensions v and w f**(v,w) = vp(w(f)), we define a symmetric bilinear f at p. If to vector fields. v,w c TMp We let * is, of course, just v. We must show that VP this is symmetric and well-defined. It is symmetric because where vp(w(f)) - wp(v(f)) _ [v,w]p(f) = 0 where [v,wl Here w(f) is the Poisson bracket of and w, denotes the directional derivative of and where [v,w] (f) = 0 f in the direction w. DEFINITIONS AND LEMMAS §2. since p has f as a critical point. f** Therefore is independent of w = E bj a-.Ip of while v, w. a is a local coordinate system and (x1,...,xn) If It is now clearly well-defined since is symmetric. is independent of the extension v vp(w(f)) = v(w(f)) wp (v(f)) 5 w = E bj aj we can take where bj v = E a. p, 1 axi now denotes a con- ax ax Then stant function. f**(v,w) = v(w(f))(p) = v(E b af) = respect to the basis I aX p ,..., a Z ij i axj f i b i ax (p) a axn p We can now talk about the index and the nullity of the bilinear functional tor space ff* on The index of a bilinear functional TMp. on a vec- H, is defined to be the maximal dimension of a subspace of V V, on which H is negative definite; the nullity is the dimension of the nullspace, i.e., the subspace consisting of all for every w e V. point of f index of f** The point if and only if on TMp p f** v E V such that H(v,w) = 0 is obviously a non-degenerate critical on TMp has nullity equal to will be referred to simply as the index of The Lemma of Morse shows that the behaviour of described by this index. at f p The 0. f Before stating this lemma we first prove the following: LEMMA 2.1. Let borhood V of f 0 be a C°° function in a convex neigh- in Rn, with n f(x1,...,xn) _ f(o) = 0. Then xigi(x1,...,xn) i=t for some suitable C" functions defined in V, gi with gi(o) = )f (o). i PROOF: df(tx1,...,txn) f(xt,...,xn) = J f0 dt = 1 I 0 i=1 of tx1,...,txn) xi dt i 1 Therefore we can let gi(x1...,xn) = f 0 (tx1,...,txn) dt i at p. can be completely I. 6 NON-DEGENERATE FUNCTIONS Let p be a non-degenerate (Lemma of Morse). Then there is a local coordinate f. LEMMA 2.2 critical point for system (y1,...,yn) for all = 0 yi(p) f - f(p) + ... + (yn)2 is the index of where U, must be the index of (z1,...,zn), (y% 2 + (yk+1)2 - at f p. We first show that if there is any such expression for PROOF: X in a neighborhood U of p with and such that the identity (Y1)2- ... - holds throughout then i at f f, For any coordinate system p. if f(q) = f(p) - (z1(q))2- ... - (ZX(q))2 + (z11+1(9.))2 + ... + (Zn(q))2 then we have 2 if i = j < x if i = J> X 0 otherwise , -2 f (p) = az1 azj which shows that the matrix representing a f*,* , , with respect to the basis is Ip,..., azn IP Therefore there is a subspace of TMp of dimension tive definite, and a subspace V of dimension n-X definite. on which If there were a subspace of f** where l where f** f** is nega- is positive of dimension greater than TMp X were negative definite then this subspace would intersect V, which is clearly impossible. Therefore X is the index of We now show that a suitable coordinate system Obviously we can assume that p f**. (y1,...,yn) is the origin of Rn and that exists. f(p) = f(o) = 0. By 2.1 we can write xi gg(x1,...,xn) f(x1,...,xn) _ J= for (x1,...,xn) 1 in some neighborhood of 0. Since critical point: go (o) = a (0) = 0 0 is assumed to be a DEFINITIONS AND LEMMAS §2. 7 we have Therefore, applying 2.1 to the gj n gj(x1)...,xn) _ xihij(x1) ...,xn) i=1 for certain smooth functions It follows that hij. n xixjhij(x1,...,xn) f(x1) ...,xn) _ i,j=1 We can assume that and then have and hij = Fiji since we can write hij = hji, f = E xixjhij hij = '-2(hij+ hji), Moreover the matrix (hij(o)) . 2 is equal to ( 2 and hence is non-singular. (o)), ax aX7 There is a non-singular transformation of the coordinate functions which gives us the desired expression for borhood of f, in a perhaps smaller neigh- To see this we just imitate the usual diagonalization proof 0. (See for example, Birkhoff and MacLane, "A survey of for quadratic forms. The key step can be described as follows. modern algebra," p. 271.) Suppose by induction that there exist coordinates a neighborhood f throughout of U1 so that 0 + (u1)2 + ... + (uY_1)2 + where the matrices U1; a linear change in the last Let n-r+1 be a smooth, non-zero function of borhood U2 C U1 vi=ui uiujHij(u1,...,un) i,j>r of are symmetric. After (Hij(u1,...,un)) coordinates we may assume that denote the square root of g(u1,...,un) in u1, ...,un u1,...,un 1Hr,r(u1,...,un)I. This will throughout some smaller neigh- Now introduce new coordinates 0. o. Hr,r(o) vl,...,vn by fori#r vr(u1)...,un) = g(u1,...,un)[ur, + uiHir(u1,...,un)/Hr,r,(u1,...sun)]' i> r It follows from the inverse function theorem that v1, ...,vn will serve as coordinate functions within some sufficiently small neighborhood It is easily verified that f = i<r f can be expressed as + (vi)2 + i,j>r vivjHij(v1,...,vn) U3 of 0. I. 8 throughout U3. NON-DEGENERATE FUNCTIONS This completes the induction; and proves Lemma 2.2. COROLLARY 2.3 Non-degenerate critical points are isolated. Examples of degenerate critical points (for functions on R and R2) are given below, together with pictures of their graphs. (a) f(x) = x3. The origin is a degenerate critical point. (b) F(x) = e- 1/X2 Sin2(,/X) The origin is a degenerate, and non-isolated, critical point. (c) f(x,y) = x3 - 3xy2 = Real part of (o.o) (x + iy)3. is a degenerate critical point (a "monkey saddle"). §2. (d) f(x,y) = x2. DEFINITIONS AND LEMMAS The set of critical points, all of which are degenerate, is the (e) f(x,y) = x2y2. 9 x axis, which is a sub-manifold of R 2. The set of critical points, all of which are degenerate, consists of the union of the not even a sub-manifold of x and y axis, which is R2. We conclude this section with a discussion of 1-parameter groups of diffeomorphisms. The reader is referred to K. Nomizu,"Lie Groups and Differ- ential Geometry;'for more details. A 1-parameter group of diffeomorphisms of a manifold M is a map W: R x M - M C00 10 I. NON-DEGENERATE FUNCTIONS such that 1) for each a 1-parameter group pt+s = Ipt cp on M as follows. a vector field X f t,s c R we have for all Given cpt: M -+ M defined by ° onto itself, ''s of diffeomorphisms of M we define For every smooth real valued function let f((Ph(q.)) Xq(f) =h lim - o This vector field X M is a diffeomorphism of = (p(t,q) cpt(q) 2) t E R the map h - f(q) is said to generate the group p. LEMMA 2.4. A smooth vector field on M which vanishes outside of a compact set K C M generates a unique 1parameter group of diffeomorphisms of M. Given any smooth curve PROOF: t - c(t) E M it is convenient to define the velocity vector c TMc(t) by the identity (f) = h yme fc(t+hh-fc(t) (Compare §8.) Now let be a 1-parameter group of diffeomorphisms, generated by the vector field X. Then for each fixed q the curve t -" pt(9) satisfies the differential equation dcpt(q) pct with initial condition dcpt(_q) (f) rlim Q = where p = cpt(q). cpe(q) = q. Xrot(q) This is true since f (Tt+h(q)) - f(cwt(9)) lim h h o f (roh(p) ) - f(p) X(f) p 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 Variables," p. 166. Note that, in terms of local coordinates u1,...,un, the differential equa- i tion takes on the more familiar form: dam- = x1(ul,...,un), i = 1,...,n.) DEFINITIONS AND LEMMAS §2. there exists a neighborhood U and a Thus for each point of M number 11 so that the differential equation s > 0 dpt(q) - = Xcpt(q), has a unique smooth solution for = q CPO(q) q e U, It! < e. The compact set K can be covered by a finite number of such neighborhoods numbers e. e0 > 0 Let U. Setting denote the smallest of the corresponding pt(q) = q for tial equation has a unique solution q e M. for (pt(q) It, < E0 and for all Further- This solution is smooth as a function of both variables. more, it is clear that Therefore each such cpt is a diffeomorphism. be expressed as a multiple of t = k(so/2) + r Nt = 9PE0/2 ° cpE /2 0 only necessary to replace ke /2 0 is defined for all values of t. for cpt Any number Iti > so. plus a remainder e0/2 with k > 0, where the transformation 1tj,1s1,1t+s1 < eo. providing that q>s cpt+s = Wt ° It only remains to define If it follows that this differen- q # K, r with can t Irk < eo/2 set CPEo/2 ° ... is iterated by (p_E /2 ° cpe0/2 k 'Pr ° times. iterated If -k k < 0 times. it is Thus (pt 0 It is not difficult to verify that well defined, smooth, and satisfies the condition (pt+s = Tt G (ps Tt is This completes the proof of Lemma 2.4 REMARK: cannot be omitted. The hypothesis that X vanishes outside of a compact set For example let M be the open unit interval and let X be the standard vector field d _ff on M. generate any 1-parameter group of diffeomorphisms of (0,1) C R, Then X does not M. 12 NON-DEGENERATE FUNCTIONS I. Homotopy Type in Terms of Critical Values. §3. Throughout this section, if manifold f is a real valued function on a we let M, Ma = f-1(- .,a] = (p e M : f(p) < a) THEOREM 3.1. on a manifold f-1[a,b], Let . be a smooth real valued function f Let a < b and suppose that the set consisting of all p e M with a < f(p) < b, M. is compact, and contains no critical points of f. Then Ma is diffeomorphic to Mb. Furthermore, Ma is a deformation retract of Mb, so that the inclusion map Ma Mb is a homotopy equivalence. The idea of the proof is to push Mb nal trajectories of the hypersurfaces Ma down to f = constant. along the orthogo- (Compare Diagram 2.) Diagram 2. Choose a Riemannian metric on M; and let < X,Y > denote the inner product of two tangent vectors, as determined by this metric. gradient of f is the vector field grad f The on M which is characterized by the identity* <X, grad f> = X(f) (= directional derivative of for any vector field X. This vector field grad f vanishes precisely at the critical points of f. If f along X) In classical notation, in terms of local coordinates gradient has components E gij f * j au3 u1,...,un, the HOMOTOPY TYPE §3. c: R M is a curve with velocity vector grad f d - 'dE 13 note the identity d(dam p: M -+ R be a smooth function which is equal to Let 1/ < grad f, grad f > throughout the compact set f-1[a,bl; and which vanishes outside of a compact neighborhood of this set. Then the vector field X, defined by Xq = p(q) (grad f)q Hence X satisfies the conditions of Lemma 2.4. generates a 1-parameter group of diffeomorphisms CPt: M - M. q E M consider the function For fixed f-1[a,bl, lies in the set df(cpt(q)) dt _< dcpt(q) d f(cpt(q)). t If Wt(q) then , grad f > = < X, grad f > = + 1 . Thus the correspondence t - f(Wt(q.)) is linear with derivative +1 as long as f(cpt(q)) Now consider the diffeomorphism Ma diffeomorphically onto Mb. cpb_a: M lies between M. a and b. Clearly this carries This proves the first half of 3.1. Define a 1-parameter family of maps rt: Mb . Mb by rt(q) = Then m' re is the identity, and is a deformation retract of REMARK: r1 J q if f(q) < a ''t(a-f(q))(q) if a < f(q) < b is a retraction from Mb Mb. The condition that to ma. Hence This completes the proof. f-1[a,b] is compact cannot be omitted. For example Diagram 3 indicates a situation in which this set is not compact. The manifold M does not contain the point formation retract of Mb. p. Clearly Ma is not a de- 14 I. NON-DEGENERATE FUNCTIONS Diagram 3. Let f: M-+ R be a smooth function, and let p be a non-degenerate critical point with index X. Setting f(p) = c, suppose that f-1[c-e,c+el is compact, and contains no critical point of f other then p, for the set some s > 0. Then, for all sufficiently small e, Mc-e with a %-cell attached. MC+e has the homotopy type of THEOREM 3.2. The idea of the proof of this theorem is indicated in Diagram 4, for the special case of the height function on a torus. The region Mc-e = f-1(-co,c-e1 is heavily shaded. We will introduce a new function coincides with the height function borhood of p. f except that F: M -+ R which F < f in a small neigh- Thus the region F-1will consist of gether with a region H near p. In Diagram 4, shaded region. MC+E Diagram it. Mc-e to- H is the horizontally §3. HOMOTOPY TYPE Choosing a suitable cell 15 a direct argument e% C H, Mc-EU ex ing in along the horizontal lines) will show that Mc-E retract of U H. of is a deformation Finally, by applying 3.1 to the function F Mc-e we will see that region F-1[c-E,c+E1 (i.e., push- and the u H is a deformation retract This will complete the proof. Mc+E. in a neighborhood U u1,.... un Choose a coordinate system of p so that the identity f = c - (u1)2- ... - (uX)2 + (ut`+1)2+... + (un)2 holds throughout Thus the critical point U. p will have coordinates u1(p) = ... = un(p) = 0 Choose e > 0 sufficiently small so that The region (1) . points other than is compact and contains no critical f-1[c-E,c+E1 p. The image of U under the diffeomorphic (2) (u...... un): imbedding U --'Rn contains the closed ball. ((u1,...,un): Now define E (u')2 < 2E) to be the set of points in U with eX 1 ... X 2 + (u)< e and X+1(u)2+ u= ... = un = 0. The resulting situation is illustrated schematically in Diagram 5. Diagram 5. 16 NON-DEGENERATE FUNCTIONS I. The coordinate lines represent the planes u1 respectively; = ... = ux = 0 ... = un = 0 u11+1= and the circle represents the boundary of the ball of radius; and the hyperbolas represent the hypersurfaces M0-e The region and f-1(c+e). is heavily shaded; the region is heavily dotted; and the region zontal dark line through p Note that e Mc-e is attached to formation retract of X f-1[c,c+el The hori- e is precisely the boundary as required. f-1[c-e,cl is lightly dotted. represents the cell M°-e n f-1(c-e) We must prove that ex, so that Mc-eu eX e is a de- Mc+e F: M -'f R Construct a new smooth function as follows. Let µ:R-pR be a C" function satisfying the conditions µ(o) > E µ(r) -1 < where dµ µ'(r) = neighborhood . U, for r > 2e 0 µ'(r) < 0 for all F coincide with Now let r, outside of the coordinate f and let F = f - µ((u1)2+. ..+(u2 + 2(uX+1 )2+...+2(un)2 within this coordinate neighborhood. It'is easily verified that well defined smooth function throughout F is a M. It is convenient to define two functions t,11: U--i [0,oo) by _ (u1)2 + ... + (uX ) 2 {u?'+1) 2 + Then f = c - t + q; ... + (un) 2 so that: F(q) = c - (q) + q(q) - u(t(q) + 2q(q)) for all q e U. ASSERTION 1. The region F-1(-oo,c+el coincides with the region Mc+e = f-1(- oo,c+el. PROOF: Outside of the ellipsoid g + 2q < 2e the functions f and U. HOMOTOPY TYPE F coincide. 17 Within this ellipsoid we have F < f = c-g+q < c+ 2g+q < c+e This completes the proof. The critical points of ASSERTION 2. are the same as those of F f. Note that PROOF: aF Tq- 1 - 2µ'(g+2q) > 1 . Since dg + dF where the covectors and dg dq dq are simultaneously zero only at the origin, has no critical points in U it follows that F By Assertion 1 together Now consider the region F-1[c-e,c+e]. with the inequality F < f other than the origin. we see that F-1[C-e,C+e] C f-1[C-e,c+e] Therefore this region is compact. except possibly But p. F(p) =c-µ(o) <c - e Hence F It can contain no critical points of contains no critical points. F-1[c-e,c+e] . Together with 3.1 this proves the following. ASSERTION 3. The region F-1is a deformation retract of Mc+e It will be convenient to denote this region F-1(-.,c-e] Mc-e u H; where In the terminology of REMARK: described as H denotes the closure of Mc-e that the manifold-with-boundary by Mc-e Smale, the region Mc-e v H is with a "handle" attached. MC-e F-1(-co,c-El - It follows from Theorem U H is diffeomorphic to fact is important in Smale's theory of differentiable manifolds. Mc+e 3.1 This (Compare S. Smale, Generalized Poincare's conjecture in dimensions greater than four, Annals of Mathematics, Vol. 74 (1961), pp. 391-4o6.) 18 I. NON-DEGENERATE FUNCTIONS Now consider the cell e X t(q) < e, Note that eX q with consisting of all points = 0. TI(q) is contained in the "handle" H. In fact, since < 0, we have F(q) < F(p) < c-e but f(q) > c-e for q e ex. Diagram 6. The present situation is illustrated in Diagram 6. Mc-E is heavily shaded; the handle and the region F-1[c-e,c+e3 ASSERTION I+. PROOF: Mc-e H is shaded with vertical arrows; is dotted. u ex lows. rt Mc-e is a deformation retract of rt: A deformation retraction Mc-e Me-E u H - indicated schematically by the vertical arrows in Diagram 6. let The region be the identity outside of U; and define rt u u H. H is More precisely within U as fol- It in necessary to distinguish three cases as indicated in Diagram 7. CASE 1. Within the region g < e let rt correspond to the trans- formation (u1,...,un) _ (u1,...,ux,tu%+1,...,tun) . HOMOTOPY TYPE §3. 19 CASE 2 CASE 2 Diagram 7. Thus is the identity and r1 fact that each rt equality Tq_ maps r0 maps the entire region into The e into itself, follows from the in- F-1(-oo,c-el > 0. CASE 2. Within the region e < g < n + e let correspond to rt the transformation (u1,...Iun) where the number -+ (ul,...,ux,stux+l,...,Stun) is defined by st e [0,11 St = t + Thus ( 1 -t) is again the identity, and r1 hypersurface f-1(c-e). remain continuous as with that of Case 1 g g be the identity. r0 -+ e, when g stu1 Note that this definition coincides -' 0. = e. q + e < g (i.e., within MC-e) let This coincides with the preceeding definition when = q + e. This completes the proof that of maps the entire region into the The reader should verify that the functions CASE . Within the region rt 1 /2 F-1(-co,c+el. MC-eu ex is a deformation retract Together with Assertion 3, it completes the proof of Theorem 3.2. REMARK 3.3. critical points More generally suppose that there are pi,...,pk with indices x1,...,xk in f- k non-degenerate 1(c). Then a x similar proof shows that MC+e has the homotopy type of x MC-eu a 1u...u e k I. NON-DEGENERATE FUNCTIONS 20 REMARK .4. Mc the set A simple modification of the proof of 3.2 shows that is also a deformation retract of Mc+e. In fact Mc is a deformation retract of F-1which is a deformation retract of (Compare Diagram 8.) Mc-E , et` Combining this fact with 3.2 we see easily that is a deformation retract of Diagram 8: Mc Mc+e MCI. is heavily shaded, and F-1[c,c+e] is dotted. THEOREM 3.5. If f is a differentiable function on a manifold M with no degenerate critical points, and if each Ma is compact, then M has the homotopy type of a CW-complex, with one cell of dimension X for each critical point of index X. (For the definition of CW-complex see J. H. C. Whitehead, Combin- atorial Homotopy I, Bulletin of the American Mathematical Society, Vol. 55, (1949), pp. 213-245.) The proof will be based on two lemmas concerning a topological space X with a cell attached. (Whitehead) Let Wo and cp1 be homotopic maps from the sphere et' to X. Then the identity map of X extends to a homotopy equivalence LEMMA 3.6. k:Xue%-*XueX 'P0 CP 1 HOMOTOPY TYPE §3. Here k by the formulas Define PROOF: k(x) = x for x E X k(tu) = 2tu for 0 < t < 2 u E k(tu) = CP2_2t(u) for < t < 1, u E and and denotes the homotopy between 9t product of the scalar Wo with the unit vector t f: e -+ X Xu u W1 is defined by similar formulas. compositions Thus k 21 kF and fk 0 91; u. to denotes the A corresponding map ex It is now not difficult to verify that the are homotopic to the respective identity maps. is a homotopy equivalence. For further details the reader is referred to, Lemma 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 W: e1''-+ X be an attaching map. Any homotopy equivalence f: X -+ Y extends to a homotopy equivalence F X uT eX -+ Y ..f(p eX. : (Following an unpublished paper by P.Hilton.) PROOF: Define F by the conditions Let g: Y -r X FIX = Fled' = identity f be a homotopy inverse to . G: Y gfp fq) by the corresponding conditions Since gfp and define f GAY is homotopic to = W, e identity. g, it follows from 3.6 that there is a homotopy equivalence k: X e u gfp W We will first prove that the composition kGF: X u ex W is homotopic to the identity map. X u ex T 22 NON-DEGENERATE FUNCTIONS I. Let ht be a homotopy between specific definitions of k, G, and F, gf note that kGF(x) = gf(x) for x E X, kGF(tu) = 2tu for'O < t < 2 u e 11F(tu) for 2 < t < 1, u e h2-2t(P(u) Using the and the identity. The required homotopy q T : X sex X.e is now defined by the formula qT(x) ,2T u , 1 gT(tu) = F o < t < for to gT(tu) Therefore for x e X , hT(x) = 12 for h2-2t+TW(u) < t < t, u e has a left homotopy inverse. The proof that F is a homotopy equivalence will now be purely formal, based on the following. ASSERTION. If a map F right homotopy inverse R (or PROOF: L) has a left homotopy inverse then F R, is a 2-sided L and a is a homotopy equivalence; and homotopy inverse. The relations LF ti identity, FR ti identity, imply that L-' L(FR) _ (LF)R ti R. Consequently RF ti IF ti identity , which proves that R is a 2-sided inverse. The proof of Lemma 3.7 can now be completed as follows. The rela- tion kGF ti identity asserts that F has a left homotopy inverse; and a similar proof shows that G has a left homotopy inverse. Step 1. Since inverse, it follows that k(GF) identity, and (GF)k a identity. k is known to have a left §3. Step 2. inverse, it follows that G is known to have a left F has (Fk)G ti identity. F(kG) a identity, and Since F also, it follows that 23 G(Fk) a identity, and Since Step 3. HOMOTOPY TYPE kG as left inverse This completes the is a homotopy equivalence. proof of 3.7. PROOF OF THEOREM 3.5. f: M - R. values of Ma is compact. a c1,c21C31... Let The sequence Ma The set ci > a. has no cluster point since each a < c1. Suppose OW-complex. Mc+e By Theorems 3.1, 3.2, and 3.3, has % MC-cu e1 u...u e3(c) for certain maps the homotopy type of q)1,. 'J (c) T1 e (ci) is of the homotopy type of a X when be the critical c1 < c2 < c3 < ... is vacuous for Ma and that be the smallest c Let is small enough, and there is a homotopy equivalence We have assumed that there is a homotopy equivalence h: h': Ma K, Mc-e - Ma. where K is a OW-complex. Then each h' o h o Tj is homotopic by cellular approximation to a map >Vj: 6 J Then K u e u...u ei(c) Mc+e, is a OW-complex, and has the same homotopy by Lemmas 3.6, 3.7. By induction it follows that each Ma' OW-complex. K. ''j (c) `1 type as -+ (), j-1) - skeleton of has the homotopy type of a If M is compact this completes the proof. If M is not com- pact, but all critical points lie in one of the compact sets proof similar to that of Theorem 3.1 shows that the set retract of M, Ma Ma, then a is a deformation so the proof is again complete. If there are infinitely many critical points then the above construction gives us an infinite sequence of homotopy equivalences Mat C Ma2 C Ma3 C f K1 each extending the previous one. I C K2 ... f C K3 C ... , Let K denote the union of the Ki in the direct limit topology, i.e., the finest possible compatible topology, and 24 let NON-DEGENERATE FUNCTIONS I. g: M -r K be the limit map. groups in all dimensions. Then g induces isomorphisms of homotopy We need only apply Theorem 1 of Combinatorial homotopy I to conclude that g is a homotopy equivalence. [Whitehead's theorem states that if M and K are both dominated by CW-complexes, then any map M -+ K which induces isomorphisms of homotopy groups is a homotopy Certainly K is dominated by itself. equivalence. To prove that M is dominated by a CW-complex it is only necessary to consider M as a retract This completes the proof of tubular neighborhood in some Euclidean space.] of Theorem 3.5. We have also proved that each Ma has the homotopy type REMARK. of a finite CW-complex, with one cell of dimension point of index X in (Compare Remark 3.4.) Ma. This is true even if a X for each critical is a critical value. EXAMPLES §4. §4. 25 Examples. As an application of the theorems of §3 we shall prove: (Reeb). If M is a compact manifold and f is a differentiable function on M with only two critical points, both of which are non-degenerate, then M is homeomorphic to a sphere. THEOREM 4.1 PROOF: This follows from Theorem 3.1 together with the Lemma of Morse (§2.2). points. If s The two critical points must be the minimum and maximum f(p) = 0 Say that is the mimimum and is small enough then the sets closed n-cells by §2.2. But Me Ms = f_1[0,e) between M and M1-E is the maximum. f_1[1-e,11 by §3.1. f-1[1-e,1), are Thus matched It is now easy to construct a homeomorphism Sn. REMARK 1. degenerate. M1-E and 1 and is homeomorphic to M is the union of two closed n-cells, along their common boundary. f(q) = The theorem remains true even if the critical points are However, the proof is more difficult. (Compare Milnor, Differ- ential topology, in "Lectures on Modern Mathematics II," ed. by T. L:.Saaty (Wiley, 1964), pp. 165-183; Theorem 1'; or R. Rosen, A weak form of the star conjecture for manifolds, Abstract 570-28, Notices Amer. Math Soc., Vol. 7 (1960), p. 380; Lemma 1.) REMARK 2. It is not true that M must be diffeomorphic to Sn with its usual differentiable structure.(Compare: Milnor, On manifolds homeomorphic 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- 26 NON-DEGENERATE FUNCTIONS I. degenerate critical points.) As another application of the previous theorems we note that if an n-manifold has a non-degenerate function on it with only three critical points then they have index 0, n and n/2 (by Poincare duality), and the manifold has the homotopy type of an n/2-sphere with an n-cell attached. See J. Eells and N. Kuiper, Manifolds which are like projective planes, Inst. des Hautes Etudes Sci., Publ. Math. 14, 1962. Such a function exists for example on the real or complex projective plane. Let CPn be complex projective n-space. equivalence classes of EIzj12 = (n+1)-tuples of complex numbers, with (z0,...,zn) Denote the equivalence class of (z0,...,zn) by (z0:z1:...:zn). 1. Define a real valued function on CPn by the identity f f(z0:z1:...:zn) = I where We will think of CPn as c0,c1,...,cn cj1zjl2 are distinct real constants. In order to determine the critical points of following local coordinate system. with z0 0, Let Zj zo and set I Za I Then U0 be the set of consider the (zo:z1:...:zn) = x+ iyj U0 - R x1,y1,...,xn,yn: are the required coordinate functions, mapping U0 the open unit ball in R 2ri. f, diffeomorphically onto Clearly Izol2 zjI2 = xj2 + yj2 = 1 - E (xj2 + yj2) so that f = c0 j=1 throughout the coordinate neighborhood f within U. U0. Thus the only critical point of lies at the center point p0 = (1:0:0:...:0) of the coordinate system. At this point index equal to twice the number of j f with is non-degenerate; and has cj < co, Similarly one can consider other coordinate systems centered at the points p1 = (0:1:0: ...:0),...,Pn = (0:0:...:0:1) §4. It follows that index of f at EXAMPLES 27 p0,pi,...,pn are the only critical points of pk is equal to twice the number of Thus every possible even index between 0 and 2n j with f. The cj < ck. occurs exactly once. By Theorem 3.5: C Pn has the homotopy type of a CW-complex of the form e° u e2 u e4 u...u e2n It follows that the integral homology groups of CPn are given by Fii(CPn;Z) Z l 0 for i = 0,2,4,...,2n for other values of i 28 NON-DEGENERATE FUNCTIONS I. The Morse Inequalities. §5. In Morse's original treatment of this subject, Theorem 3.5 was not The relationship between the topology of M and the critical available. points of a real valued function on M were described instead in terms of a collection of inequalities. This section will describe this original point of view. DEFINITION: the integers. S S(X,Y) + S(Y,Z). Let S be a function from certain pairs of spaces to If equality holds, as coefficient group, let RX(X,Y) = Xth Betti number of = rank over (X,Y) S(X,Z) < is called additive. S As an example, given any field F for any pair we have is subadditive if whenever XD Y: )Z F (X,Y) HX(X,Y;F) of such that this rank is finite. RX is subadditive, as is easily seen by examining the following portion of the exact sequence for (X,Y,Z): HX(X,Z) - HX(X,Y) -+ ... ... -+ HX(Y,Z) The Euler characteristic X(X,Y) is additive, where X(X,Y) _ E (-1)X RX(X,Y). LEMMA 5.1. Then Let S be subadditive and let S(Xn,XO)< S(Xi,Xi_,). If S X0C...C Xn. is additive then equality holds. 1 PROOF: the case n = 2 Induction on n. For n = 1, equality holds trivially and is the definition of [sub] additivity. n-1 If the result is true for n - 1, then S(Xn_1,XO) < } Therefore S(Xn,X0) < S(Xn_1,Xa) + S(Xn,Xn_1) < 2; S(Xi,X1_1) S(Xi,Xi_1). and the result 1 is true for Let n. S(X,o) = S(X). S(Xn) < (1) with equality if Taking X. = 0 n S is additive. in Lemma 5.1, we have S(Xi,Xi-1) §5. THE MORSE INEQUALITIES Let M be a compact manifold and 29 a differentiable function f on M with isolated, non-degenerate, critical points. be such that contains exactly Mai a1 <...< ak Let critical points, and i Mak M. = Then e% 1 Mai-1) = H (Mai-1 H*(Mat Mat-1) where is the index of the critical point, X. ), . H*(e 1,e 1) by excision, coefficient group in dimension otherwise. 0 Applying (1) to 0 = Mo a C...C M ak = M with k Xi S = R. we have R),(Ma1,Ma1-1) = CX; i=1 where C. denotes the number of critical points of index formula to the case S = X X. Applying this we have k X(Ma1,Ma1-1) x(M) = = C0 - C1 + C2 -+...+ Cn i=1 Thus we have proven: THEOREM 5.2 (Weak Morse Inequalities). If Cx denotes the number of critical points of index X on the compact mani- fold M then RX(M) < CX , and (-1)X CX E (-1)" RX(M) Slightly sharper inequalities can be proven by the following argument. LEMMA 5.3. The function S. is subadditive, where SX(X,Y) = RX(X,Y) - Rx_1(X,Y) + Rx_2(X,Y) - +...+ R0(X,Y) PROOF: Given an exact sequence h+A' B Ck... ...-+D-+0 of vector spaces note that the rank of the homomorphism h plus the rank of i is equal to the rank of A. Therefore, 30 NON-DEGENERATE FUNCTIONS I. rank h = rank A - rank i = rank A - rank B + rank j = rank A - rank B + rank C = rank A - rank B + rank C - +...+ rank D Hence the last expression is > 0. - rank k . Now consider the homology exact sequence Applying this computation to the homomorphism of a triple X J Y J Z. Hx+1 (X,Y) a H,,(Y,Z) we see that rank 3 = RX(Y,Z) - RX(X,Z) + RX(X,Y) - Rx_1(Y,Z) + ... > o Collecting terms, this means that S,(Y,Z) - S (X,Z) + SX(X,Y) > o which completes the proof. Applying this subadditive function S1, to the spaces 0 C Mat C Mat C...C Mak we obtain the Morse inequalities: k X(Mi'Mi-1) a a = CX - CX-1 +-...+ Co Sx(M) < i=1 or (40 R>(M) - RX_1(M) +-...+ Ro(M) < CX - CX_1+ -...+ Co. These inequalities are definitely sharper than the previous ones. In fact, adding with (4,_1) (4x) and (4,_1), one obtains (2x); for x> n one obtains the equality and comparing (40 (3). As an illustration of the use of the Morse inequalities, suppose that C,+1 = 0. (4.) and Then R,+1 (41,+1), must also be zero. we see that RX - R)6_1 +-...+ Ro Now suppose that Comparing the inequalities C.-1 = is also zero. C% - C1'_1 -...± Co Then R._1 = 0, . and a similar argu- ment shows that Rx-2 - Rx-3 +-...± Ro = Cx-2 - Cx-3 +-...± Co . §5. THE MORSE INEQUALITIES 31 Subtracting this from the equality above we obtain the following: COROLLARY 5.4. If Cx+1 = CX_1 = 0 then R. = C. and R)L+1 = RX_1 = 0. (Of course this would also follow from Theorem 3.5.) Note that this corollary enables us to find the homology groups of complex projective space (see §4) without making use of Theorem 3.5. 32 NON-DEGENERATE FUNCTIONS I. §6. Manifolds in Euclidean Space. Although we have so far considered, on a manifold, only functions which have no degenerate critical points, we have not yet even shown that In this section we will construct many func- such functions always exist. tions with no degenerate critical points, on any manifold embedded in Rn. In fact, if for fixed p c Rn define the function lip-gII2. It will turn out that for almost all LP: M - R by LL(q) _ the function Lp p, has only non-degenerate critical points. Let M C Rn be a manifold of dimension bedded in Rn. Let N = ((q,v): q e M, v differentiably embedded in R2ri. Let differentiably em- perpendicular to M at It is not difficult to show that N vector bundle of k < n, N C M x Rn be defined by (N q). is an n-dimensional manifold is the total space of the normal M.) E: N -+ Rn be E(q,v) = q + v. (E is the "endpoint" map.) E(4,v) DEFINITION. µ if nullity e = q + v µ > 0. e E Rn is a focal point of where (q,v) E N and the Jacobian of The point a focal point of (M,q) e with multiplicity (M,q) E at (q,v) will be called a focal point of M if for some has e q e M. Intuitively, a focal point of M is a point in Rn where nearby normals intersect. We will use the following theorem, which we will not prove. is §6. MANIFOLDS IN EUCLIDEAN SPACE 33 THEOREM 6.1 (Sard). If M1 and M2 are differentiable manifolds having a countable basis, of the same dimension, then the image of the M2 is of class C1, and f: M1 set of critical points has measure A critical point of in 0 M2- is a point where the Jacobian of f f is For a proof see de Rham, "Variotes Differentiables," Hermann, singular. Paris, 1955, p. 10. COROLLARY 6.2. PROOF: E: N -s Rn. is M. N We have just seen that is a focal point iff x x the point For almost all x e Rn, not a focal point of is an n-manifold. The point x is in the image of the set of critical points of Therefore the set of focal points has measure 0. For a better understanding of the concept of focal point, it is convenient to introduce the "second fundamental form" of a manifold in Euclidean space. We will not attempt to give an invariant definition; but will make use of a fixed local coordinate system. Let u1,...,uk be coordinates for a region of the manifold M C R. Then the inclusion map from M to Rn determines n smooth functions k x1(u 1 ,...,uk ),...,xn(u '...,u) 1 These functions will be written briefly as x(u1,...,uk) (x1,...,xn). To be consistent the point where x = q e M C Rn will now be denoted by q. The first fundamental form associated with the coordinate system is defined to be the symmetric matrix of real valued functions (gij) The second fundamental form (3ui _ j) - . on the other hand, is a symmetric matrix (ij) of vector valued functions. 2-' at a point of M can The vector au aukk be expressed as the sum of a vector tangent to M and a vector normal to M. It is defined as follows. 2- Define T.. x to be the normal component of au au v which is normal to M at q the matrix (v auJ ) _ (v i.1j . Given any unit vector 34 I. NON-DEGENERATE FUNCTIONS at q in the direction M can be called the "second fundamental form of v." It will simplify the discussion to assume that the coordinates have been chosen to that evaluated at q, gi3, Then the eigenvalues of the matrix curvatures K1,...,KK of ( v M at -1 is the identity matrix. are called the principal in the normal direction v. The re- of these principal curvatures are called the princi- ciprocals pal radii of curvature. is singular. Of course it may happen that the matrix (v .7 ) Ki will be zero; and hence In this case one or more of the the corresponding radii ii will not be defined. Kit Now consider the normal line where consisting of all q + tv, 2 M at q v is a fixed unit vector orthogonal to LEMMA 6.3. The focal points of (M q) along f are precisely the points _q q+ Ki-1 _V, where 1 < i< k, Ki / o. v Thus there are at most, k focal points of (M,q) along f, each being counted with its proper multiplicity. Choose PROOF: along the manifold so that w1,...;wn_k nal to each other and to t1,...Itn-k) to-k) k 1 n-k vector fields are unit vectors which are orthogo- We can introduce coordinates M. N C M x e as follows. on the manifold k 1 w1(u ,...,u ),...,wn_k(u ,...,u ) Let (u1,...,uk, (u1,...,uk,t1,.. correspond to the point n-k \ to wa(u1,...,uk)1 e N (x(u1,...,uk), a=1 Then the function -. Rn E:N gives rise to the correspondence (ll ,...,U ,t ,...,t n-k ) 1 k 1 e' 1 k X(u ,...,u ) + J a -+ 1 k wa(u ,...,u ) t , with partial derivatives aB aX aul au1 + to awa aul ae a a Taking the inner products of these n-vectors with the linearly independent MANIFOLDS IN EUCLIDEAN SPACE §6. vectors axl, axk, w1'...;wn_k we will obtain an nxn matrix whose au au rank equals the rank of the Jacobian of This 35 E at the corresponding point. nxn matrix clearly has the following form ax au aX y awa + to auj ax w 1 1 to awa \1 aul 0 / ( TIT a S identity 0 matrix Thus the nullity is equal to the nullity of the upper left hand block. Using the identity o= (W 7'T 1 ax a = auk aWa ax au1 auk x + Wa 2)u1auj we see that this upper left hand block is just the matrix -. ta wa ( gig Iii a Thus: ASSERTION 6.4. q + tv µ is a focal point of (M,q) with multiplicity if and only if the matrix (gig - tv Iii ) (*) is singular, with nullity Now suppose that lar if and only if -f more the multiplicity (gij) P. is the identity matrix. is an eigenvalue of the matrix (v µ Then (*) is singu) Further- is equal to the multiplicity of - as eigenvalue. This completes the proof of Lemma 6.3. Now for fixed p E Rn let us study the function Ip = f: M-R where f(x(u1,...,uk)) = Ij-X(u1,...,uk) - pII2 = x x - 2x p + p p . We have of aul Thus at f q = 2 ax TIT (x - p) has a critical point at q if and only if q - p is normal to M 36 NON-DEGENERATE FUNCTIONS I. The second partial derivatives at a critical point are given by 2f_ 2( ax aul = au au Setting p = x + tv, , ax x + auk . lX 3 a as in the proof of Lemma 6.3, this becomes 2(gi. - tv au auk Therefore: The point q e M is a degenerate critical point f = L5- if and only if p is a focal point of (M,q). The nullity of q as critical point is equal to the multiplicity of p as focal point. LEMMA 6.5. of Combining this result with Corollary 6.2 to Sari's theorem, we immediately obtain: (all but a set of THEOREM 6.6. For almost all p e Rn measure 0) the function LL: M -+R has no degenerate critical points. This theorem has several interesting consequences. there exists a dif- On any manifold M COROLLARY 6.7. ferentiable function, with no degenerate critical points, for which each Ma PROOF: is compact. This follows from Theorem 6.6 and the fact that an n-dimen- sional manifold M can be embedded differentiably as.a closed subset of R2n+1 (see Whitney, Geometric Integration Theory, p. 113). A differentiable manifold has the homotopy type of APPLICATION 1. a CW-complex. This follows from the above corollary and Theorem 3.5. On a compact manifold M APPLICATION 2. X there is a vector field such that the sum of the indices of the critical points of X x(M), the Euler characteristic of any differentiable function f M. This can be seen as follows: on M we have is the number of critical points with index the vector field grad f equals at a point where X. f x(M) = E (-1)x Cx But (-1)x has index x. for where C. is the index of MANIFOLDS IN EUCLIDEAN SPACE §6. 37 It follows that the sum of the indices of any vector field on M is equal to because this sum is a topological invariant x(M) (see Steen- rod, "The Topology of Fibre Bundles," §39.7). The preceding corollary can be sharpened as follows. Let k > 0 be an integer and let K C M be a compact set. COROLLARY 6.8. Any bounded smooth function f: M -+ R can g which be uniformly approximated by a smooth function has no degenerate critical points. Furthermore g can be chosen so that the i-th derivatives of g on the compact set K uniformly approximate the corresponding derivatives of f, for i < k. (Compare M. Morse, The critical points of a function of n vari- ables, Transactions of the American Mathematical Society, Vol. 33 (1931), pp. 71-91.) PROOF: Choose some imbedding h: M -+ Rn of M as a bounded sub- set of some euclidean space so that the first coordinate the given function f. Let c be a large number. p = (-C+e1,e2,..., En) (-C,0,...,o) E Rn close to degenerate; so that the function g (x) is non-degenerate. c is large and the C2 2C 1 f M -+ R is non- LP: A short computation shows that g(x) = f(x) + I hi(x)2/2c - Clearly, if Choose a point and set g(x) Clearly is precisely h1 ei ei2/2c eihi(x)/c + 1 1 are small, then g will approximate as required. The above theory can also be used to describe the index of the function at a critical point. LEMMA 6.9. (Index theorem for Lp.) The index of Lp at a non-degenerate critical point q e M is equal to the number of focal points of (M,q) which lie on the segment from q to p; each focal point being counted with its multiplicity. 38 I. NON-DEGENERATE FUNCTIONS An analogous statement in Part III (the Morse Index Theorem) will be of fundamental importance. The index of the matrix PROOF: a2 au au 2(gij - tv is equal to the number of negative eigenvalues. ij) Assuming that (gij) the identity matrix, this is equal to the number of eigenvalues of (v which are > . 7 is lid) Comparing this statement with 6.3, the conclusion follows. THE LEFSCHETZ THEOREM §7. 39 The Lefschetz Theorem on Hyperplane Sections. §7. As an application of the ideas which have been developed, we will prove some results concerning the topology of algebraic varieties. These were originally proved by Lefschetz, using quite different arguments. The present version is due to Andreotti and Frankel*. THEOREM 7.1. If M C Cn is a non-singular affine algebraic variety in complex n-space with real dimension 2k, then for Hi(M;Z) = 0 i > k. This is a consequence of the stronger: A complex analytic manifold M of complex bianalytically embedded as a closed subset of Cn has the homotopy type of a k-dimensional CW-complex. THEOREM 7.2. dimension k, The proof will be broken up into several steps. quadratic form in k Q(z1,...,zk) = I bhj If we substitute xh + iyh for obtain a real quadratic form in zh, zhzi ASSERTION 1. -e If e . and then take the real part of bhj(xh+iyh)(x3+iy3) is an eigenvalue of Q' with multiplicity is also an eigenvalue with the same multiplicity PROOF. The identity the quadratic form change of variables. Q' Q we 2k real variables: Q'(x1,...,xk,yt,...,yk) = real part of then First consider a complex variables Q(iz1,...,izk) = can be transformed into µ. -Q(z1,...,zk) -Q' µ, shows that by an orthogonal Assertion 1 clearly follows. 11 See S. Lefschetz, "L'analysis situs et la geometrie algebrique," Paris, and A. Andreotti and T. Frankel, The Lefschetz theorem on hyperplane sections, Annals of Mathematics, Vol. 69 (1959), pp. 713-717. 1921+; 4o NON-DEGENERATE FUNCTIONS I. Now consider a complex manifold M which is bianalytically imbedded as a subset of Cn. Let be a point of q The focal points of ASSERTION 2. M. along any normal line (M,q) occur in pairs which are situated symmetrically about In other words if q + tv 2 q. then q - tv is a focal point, is a focal point with the same multiplicity. PROOF. Choose complex coordinates z1(q) = ... = zk(q) borhood of q so that determines n complex analytic functions Let The inclusion map M-+ Cn = 0. a = 1, wa(z1 ,...,zk ), WC, for M in a neigh- z1,...,zk ..,n. be a fixed unit vector which is orthogonal to M at v q. Consider the Hermitian inner product wa(z1,...,zk)va wava = of w and v. This can be expanded as a complex power series I wa(z1,...,zk)va = constant + Q(z1,...,zk) + higher terms, where Q denotes a homogeneous quadratic function. ish since v is orthogonal to Now substitute system for M; (The linear terms van- M.) xh + iyh so as to obtain a real coordinate zh for and consider the real inner product w v = real part of wava . This function has the real power series expansion xk w- v = constant + Q1(x1 Clearly the quadratic terms M Q' at q v. occur in equal and opposite pairs. and k higher terms. determine the second fundamental form of Q' in the normal direction along the line through q 1 ,Y q + v By Assertion 1 the eigenvalues of Hence the focal points of (M,q) also occur in symmetric pairs. This proves Assertion 2. We are now ready to prove 7.2. squared-distance function Choose a point p e Cn so that the THE LEFSCHETZ THEOREM §7. 41 Y M -+ R Since M is a closed subset of Cn, has no degenerate critical points. it is clear that each set Ma = is compact. Now consider the index of at a critical point Lp ing to 6.9, this index is equal to the number of focal points of which lie on the line segment from p to buted symmetrically about and q. (M,q) 2k But there are at most q. and focal points along the full line through p Accord- q. Hence at most k and these are distri- q; of them can lie between p q. Thus the index of Lp at q is It follows that M has the < k. which completes the proof homotopy type of a OW-complex of dimension < k; of 7.2. COROLLARY 7.3 Let V be an algebraic variety (Lefschetz). k which lies in the complex projective space CPn. Let P be a hyperplane in CPn which contains the singular points (if any) of V. Then the inclusion map of complex dimension V n P-V induces isomorphisms of homology groups in dimensions less than Furthermore, the induced homomorphism k-1. Hk_1(V n P;Z) Hk-l(V;Z) is onto. PROOF. Using the exact sequence of the pair clearly sufficient to show that H1,(V,V n P;Z) = 0 for (V,V n P) it is r < k-1. But the Lefschetz duality theorem asserts that Hr,(V,V n P;Z) = But V -(V n P) CPn - P. H2k-r(V -(V n P) ;Z) is a non-singular algebraic variety in the affine space Hence it follows from 7.2 that the last group is zero for This result can be sharpened as follows: THEOREM 7.4 (Lefschetz). Under the hypothesis of the preceding corollary, the relative homotopy group Trr(V,V n P) is zero for r < k. r < k-1. 42 I. PROOF. borhood U NON-DEGENERATE FUNCTIONS The proof will be based on the hypothesis that some neigh- of V n P can be deformed into V n P within V. This can be proved, for example, using the theorem that algebraic varieties can be triangulated. In place of the function where LP: V - V n P -+ R we will use ro f(x) for x e V n P t/Lp(x) Since the critical points of the critical points of homotopy type of Lp f have index > no degenerate critical points with Ve = f-1[o,e] f: V -+ R for x j P. < k it follows that have index 2k - k = k. s < f < co. The function f has Therefore V has the with finitely many cells of dimension > k attached. Choose r-cube. e small enough so that Then every map of the pair VE C U. (Ir,Ir) Let into Ir denote the unit (V,V n P) can be deform- ed into a map (Ir,Ir) - (V-,V n P) C (U,V n P) since proof. r < k, and hence can be deformed into V n P. , This completes the PART II A RAPID COURSE IN RIEMANNIAN GEOMETRY §8. Covariant Differentiation The object of Part II will be to give a rapid outline of some basic concepts of Riemannian geometry which will be needed later. For more infor- mation the reader should consult Nomizu, "Lie groups and differential geoMath. Soc. Japan, 1956; Helgason, "Differential geometry and sym- metry. metric spaces," Academic Press, 1962; Sternberg, "Lectures on differential geometry," Prentice-Hall, 1964; or Laugwitz, "Differential and Riemannian geometry," Academic Press, 1965. Let M be a smooth manifold. DEFINITION. p E M An affine connection at a point which assigns to each tangent vector Xp E TMp is a function and to each vector field Y a new tangent vector Xp - Y E TMp of Y in the direction Xp. called the covariant derivative quired to be bilinear as a function of Xp and Y. This is re- Furthermore, if f: M- R is a real valued function, and if fY denotes the vector field (fY)q = f(q)Yq 'then h is required to satisfy the identity Xp I- (fY) = (Xpf)Yp + f(p)Xp I- Y . Note that our X F Y coincides with Nomizu's Vxy. The notation is intended to suggest that the differential operator X acts on the vector field Y. 44 RIEMANNIAN GEOMETRY II. (As usual, denotes the directional derivative of Xpf in the direction f XP.) (or briefly a connection) on M is a A global affine connection function which assigns to each p e M an affine connection at Fp p, satisfying the following smoothness condition. and Y are smooth vector fields on M X If 1) then the vector defined by the identity field X F Y, (X FY)p=Xp FpY , must also be smooth. Note that: (2) X F Y is bilinear as a function of (3) (fX) (4) (X F (fY) = (Xf)Y + f(X F Y) F Y = f(X F Y) and Y X , Conditions (1), (2), (3), (4) can be taken as the definition of a connection. In terms of local coordinates neighborhood U C M, the connection valued functions ik field on U. on ul,...,un is determined by F as follows. U, defined on a coordinate Then any vector field X Let n3 smooth real ak denote the vector on U can be expressed au uniquely as X = k xkak k=1 where the field (5) xk ai F ai are real valued functions on U. In particular the vector can be expressed as ai F a _ rij k These functions i, determine the connection completely on In fact given vector fields X x'ai = expand X F Y by the rules (2), (3), (4); (6) XFY = k i and Y = yJaj one can yielding the formula xiyki)ak U. §8. where the symbol yki 45 COVARIANT DIFFERENTIATION stands for the real valued function y'i aiyk + = ij y j Conversely, given any smooth real valued functions ik one can define X F Y by the formula (6). the conditions The result clearly satisfies one can define the covariant derivative of F a vector field along a curve in First some definitions. M. A parametrized curve in M is a smooth function M. assigns to each U, (1), (2), (3), (4), (5). Using the connection numbers to on A vector field V along the curve c c from the real is a function which t e R a tangent vector Vt E TMc(t) This is required to be smooth in the following sense: tion For any smooth func- on M the correspondence f t tf V should define a smooth function on R. As an example the velocity vector field vector field along c do Tdt of the curve is the which is defined by the rule UE Here (if - c* d T£ denotes the standard vector field on the real numbers, and c*: TRt TMc(t) denotes the homomorphism of tangent spaces induced by the map Diagram 9.) de Diagram 9 c. (Compare 46 II. RIEMANNIAN GEOMETRY Now suppose that M is provided with an affine connection. any vector field V along determines a new vector field c called the covariant derivative of DV Then along c The operation V. DV ffu V is characterized by the following three axioms. a) D(V+W) Tf- = ur+D(fV) c) V+f _ DV . If V is induced by a vector field Y on Vt Yc(t) for each then t, DV that is if M, is equal to _CTE F Y (= the covariant derivative of Y in the direction of the velocity vector of c) There is one and only one operation V -+ DV which satisfies these three conditions. LEMMA 8.1. PROOF: u1(t),. .,un(t) field V Choose a local coordinate system for denote the coordinates of the point and let The vector c(t). can be expressed uniquely in the form V where M, v1,...,vn subset of R), I v4j = are real valued functions on R and (or an appropriate open a1,...1an are the standard vector fields on the co- ordinate neighborhood. It follows from (a), (b), and (c) that i dam- k = ri vi, ak + 3 i,j Conversely, defining DV by this formula, it is not difficult to verify that conditions (a), (b), and (c) A vector field V along if the covariant derivative CT:E are satisfied. c is said to be a parallel vector field is identically zero. §8. 47 COVARIANT DIFFERENTIATION Given a curve c and a tangent vector V0 there is one and only one parallel c(o), vector field V along c which extends Vo. LEMMA 8.2. at the point The differential equations PROOF. dv k du' r k vi ij + i,j have solutions vk(o). 0 = which are uniquely determined by the initial values vk(t) Since these equations are linear, the solutions can be defined for all relevant values of p. 152.) Real Variables," The vector Vt lation along (Compare Graves, "The Theory of Functions of t. is said to be obtained from V0 by parallel trans- c. Now suppose that M Xp, Yp will be denoted by of two vectors DEFINITION. The inner product is a Riemannian manifold. A connection < Xp, Yp > . on M is compatible with the Rieman- F nian metric if parallel translation preserves inner products. for any parametrized curve along c, and any pair c P, P' In other words, of parallel vector fields the inner product < P,P' > should be constant. Suppose that the connection is compatible with LEMMA 8.3. Let V, W be any two vector fields along the metric. c. Then E < V,W > _ <Ur,W> <V, UE> + Choose parallel vector fields PROOF: are orthonormal at one point of c v1 lows that < V,W > _ I v1Pi >, v'w' DW T£ dv1 Therefore + <VrUT which completes the proof. _ which c Then c. wiPj and respecIt fol- and that Pi, _dT ,W> along = < V,Pi > is a real valued function on R). DV < P1,...,Pn and hence at every point of the given fields V and W can be expressed as tively (where . d i dwJ _ wi + v1 d i\ ) P = d <V,W> 48 RIEMANNIAN GEOMETRY II. For any vector fields vector XP e TMp: COROLLARY 8.4. Xp <Y,Y' > Choose a curve PROOF. <Xp F Y,YY> _ c on M and any Y,Y' + <Yp,Xp F Y' > . whose velocity vector at t = 0 is Xp; and apply 8.3. A DEFINITION 8.5. connection is called symmetric if it satis- F fies the identity* (X F Y) - (Y F X) (As usual, [X,Y] denotes the poison bracket two vector fields.) since = [ai,ajl = 0 [X,Y] . [X,Y]f = X(Yf) - Y(Xf) Applying this identity to the case X = ai, of Y = aJ, one obtains the relation rij - rjk Conversely if ik = rji = o. then using formula (6) it is not difficult to verify that the connection F is symmetric throughout the coordinate neigh- borhood. LEMMA 8.6. (Fundamental lemma of Riemannian geometry.) A Riemannian manifold possesses one and only one symmetric connection which is compatible with its metric. (Compare Nomizu p. 76, Laugwitz p. 95.) PROOF of uniqueness. and setting Permuting Applying 8.4 to the vector fields < aj,ak > = gjk one obtains the iaentity ai gjk = < ai F apak > + < aj,ai F ak > i,j, and k ai,a3,ak, . this gives three linear equations relating the * The following reformulation may (or may not) seem more intuitive. Define The "covariant second derivative" of a real valued function f along two vectors XpYp to be the expression Xp(Yf) - (Xp F Y)f where Y denotes any vector field extending Yp. It can be verified that this expression does not depend on the choice of Y. (Compare the proof of Lemma 9.1 below.) Then the connection is symmetric if this second derivative is symmetric as a function of Xp and Yp. COVARIANT DIFFERENTIATION §8. 49 three quantities < i F aj,akKaj aj> k'ai>, and <ak F (There are only three such quantities since ai F 3j = F ai aj .) These equations can be solved uniquely; yielding the first Christoffel identity ai F aj,ak > = 2 (aigjk + ajgik - akgij) The left hand side of this identity is equal to rij gkk . Multiplying . P by the inverse (g' ) of the matrix this yields the second Christof- (gQk) fel identity r j 2 \ai gjk + aj gik - ak gij) gk¢ = k Thus the connection is uniquely determined by the metric. Conversely, defining by this formula, one can verify that the rl resulting connection is symmetric and compatible with the metric. This completes the proof. An alternative characterization of symmetry will be very useful Consider a later. "parametrized surface" in s: By a vector field V along (x,y) a R2 M: that is a smooth function R2 - M is meant a function which assigns to each s a tangent vector V(x,Y) c TM5(x,Y) As examples, the two standard vector fields tor fields s, . Jx and y ; and along s* -jV s. yo, give rise to vec- and called the "velocity vector fields" of For any smooth vector field V along and and TV x These will be denoted briefly by s are new vector fields, constructed as follows. Ty- s. the covariant derivatives For each fixed restricting V to the curve x s(x,Y0) one obtains a vector field along this curve. respect to x is defined to be ()(x the entire parametrized surface Y) 'o Its covariant derivative with This defines -DV along s. As examples, we can form the two covariant derivatives of the two II. 50 vector fields as cTx and as 3y RIEMANNIAN GEOMETRY The derivatives simply the acceleration vectors of suitable and the mixed derivatives LEMMA 8.7. PROOF. and compute. If D D as Tx and D as are cTy cTy coordinate curves. However, cannot be described so simply. the connection is symmetric then D as ax ay = D as ay ax Express both sides in terms of a local coordinate system, §9. THE CURVATURE TENSOR §9. The Curvature Tensor The curvature tensor R of an affine connection extent to which the second covariant derivative metric in and i Given vector fields j. 51 measures the F ai F () j F Z) is sym- define a new vector field X,Y,Z R(X,Y)Z by the identity* R(X,Y) Z = -X F (Y F Z) + Y F (X F Z) + [X,Y1 F Z The value of R(X,Y)Z at a point p E M depends only on the vectors Xp,Yp,Zp at this point p and not on their values at nearby points. Furthermore the correspondence R(Xp,Yp)Zp Xp,Yp,Zp LEMMA 9.1. from to TMp x TMp x TMp is tri-linear. TMp Briefly, this lemma can be expressed by saying that R is a "tensor." PROOF: If X Clearly R(X,Y)Z is replaced by a multiple is a tri-linear function of then the three terms fX X,Y, and Z. -X F (Y F Z), Y F (X F Z), [X,Y] F Z are replaced respectively by i) - fX F (Y F Z) , ii) iii) (Yf) (X F Z) + fY F (X F Z) - (Yf)(X F Z) + f[X,Y1 F 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 xiai, = Y = y'aj , and Z = zk)k. Then R(X,Y)Z = R(xiai,yjaj)(zk) k) i i k x y z R(ai)aj)ak * Nomizu gives R the opposite sign. Our sign convention has the advantage that (in the Riemannian case) the inner product < Ro hl)i)aj,ak coincides with the classical symbol Rhijk 52 RIEMANNIAN GEOMETRY II. p Evaluating this expression at (R(X,Y)Z)p one obtains the formula x1(p)yj(p)zk(p)(R(ai,aj)ak)p = xi,yJ,zk at which depends only on the values of the functions and p, This completes the proof. not on their values at nearby points. Now consider a parametrized surface s: R2,M Given any vector field V along one can apply the two covariant dif- s. ferentiation operators -& and D to In general these operators will V. not commute with each other. LEMMA 9.2. - R( TX_,Ty) V Tx rTy V cTy 3x V Express both sides in terms of a local coordinate system, PROOF: and compute, making use of the identity a1 F (ai F ak) - ai F (ai F ak) R(ai,ai )ak = . [It is interesting to ask whether one can construct a vector field P along which is parallel, in the sense that s P = - P 0, = _5_X_ and which has a given value at the origin. P(0 0) In general no such , vector field exists. then P can be constructed as follows. field along the fixed x0 However, if the curvature tensor happens to be zero Let For each be a parallel vector field along the curve let P(xo,y) having the right value for P be a parallel vector x-axis, satisfying the given initial condition. y - s(x0,y) Clearly P(x,o) y = 0. , This defines and ZX P is identically zero; P everywhere along Now the identity D D 3yTxP implies that y zx- P = 0. D D TxF = as as R( \3x'T P = 0 In other words, the vector field D P parallel along the curves y s(x0,y) s. is zero along the x-axis. is THE CURVATURE TENSOR §9. Since (3Dx P)(x o 0) = this implies that 0, and completes the proof that P - 53 x P is identically zero; is parallel along Henceforth we will assume that s.] M is a Riemannian manifold, pro- vided with the unique symmetric connection which is compatible with its metric. In conclusion we will prove that the tensor satisfies four R symmetry relations. The curvature tensor of a Riemannian manifold LEMMA 9.3. satisfies: (1) R(X,Y)Z + R(Y,X)Z = 0 (2) (3) (4) R(X,Y)Z + R(Y,Z)X + R(Z,X)Y = 0 <R(X,Y)Z,W> + <R(X,Y)W,Z> = 0 <R(X,Y)Z,W> _ <R(Z,W)X,Y> PROOF: definition of The skew-symmetry relation (1) follows immediately from the R. Since all three terms of (2) are tensors, it is sufficient to prove (2) when the bracket products zero. and [X,Y], [X,ZI [Y,Z1 are all Under this hypothesis we must verify the identity X F (Y F Z) + Y F (X F Z) Y F (Z F X) Z F (X FY) + Z F (Y F X) + X F (Z FY) = o . But the symmetry of the connection implies that Y FZ - Z FY = [Y,ZI = 0 . Thus the upper left term cancels the lower right term. maining terms cancel in pairs. Similarly the re- This proves (2). To prove (3) we must show that the expression < R(X,Y)Z,W > skew-symmetric in Z and W. that <R(X,Y)Z,Z> for all X,Y,Z. R(X,Y)Z,Z > is This is clearly equivalent to the assertion Again we may assume that = 0 [X,YI = so that 0, is equal to - X F (Y F Z) + Y F (X F Z),Z> . 54 II. RIEMANNIAN GEOMETRY In other words we must prove that the expression <Y F (X F Z),Z> is symmetric in X Since and Y. [X,Y] and = Y. 0 the expression YX < Z,Z > is symmetric in X Since the connection is compatible with the metric, we have X < Z,Z > 2 <X F Z,Z > = hence YX <Z,Z> = 2 <Y F(X F Z),Z> + 2 <X F Z,Y F Z> But the right hand term is clearly symmetric in X Y F (X F Z),Z > is symmetric in X and and Y. Therefore Y; which proves property (3). Property (4) may be proved from (1), (2), and (3) as follows. <R(X,Y)Z,W> <R (Z,W)X,Y> 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 these identities for the top two shaded triangles, and subtracting the identities for the bottom ones, this means that twice the top vertex minus twice the bottom vertex is zero. This proves (4), and completes the proof. §10. GEODESICS AND COMPLETENESS §10. Geodesics and Completeness 55 Let M be a connected Riemannian manifold. A parametrized path DEFINITION. I y: where I -+ M, acceleration vector field UE identity d£ shown that the length constant along y. af, uu Thus the velocity is identically zero. Tdf , 2 < UE = H , dy _ 0 of the velocity vector is _H Introducing the arc-length function s(t) = \ DD IIdt + constant This statement can be rephrased as follows: The parameter geodesic is a linear function of the arc-length. ally equal to the arc-length if and only if Idyll t The parameter = t-+ y(t) e M determines The equation n along a t is actu- 1. In terms of a local coordinate system with coordinates a curve if the denotes any interval of real numbers, is called a geodesic smooth functions u1,...,un u1(t),...,un(t). -aE UE for a geodesic then takes the form d2uk + r k un) dul duJ (u1 - o i,J=1 The existence of geodesics depends, therefore, on the solutions of a certain system of second order differential equations. More generally consider any system of equations of the form -11 = dt2 Here u stands for (ul,...,un) and FI stands for an n-tuple of functions, all defined throughout some neighborhood U of a point (u1,v1) a R en C°° 56 II. RIEMANNIAN GEOMETRY There exists a EXISTENCE AND UNIQUENESS THEOREM 10.1. and a number neighborhood W of the point (u1,-v1) e > o so that, for each (u0V0) E W the differential equation dt2(u' at) has a unique solution t -; u(t) which is defined for and satisfies the initial conditions Iti < e, y u0, u(0) v0 - (0) Furthermore, the solution depends smoothly on the initial conditions. In other words, the correspondence (u0,v0,t) -' u(t) from W x (-e,e) to is a R n C°° function of all 2n+1 variables. i PROOF: vi = 3T this system of Introducing the new variables second order equations becomes a system of n 2n first order equations: du = dv _df V =(u,V) The assertion then follows from Graves, "Theory of Functions of Real Variables," p. 166. (Compare our §2.4.) Applying this theorem to the differential equation for geodesics, one obtains the following. For every point p0 on a Riemannian M there exists a neighborhood U of p0 and a number e > 0 so that: for each p c U and each tangent vector v E TMp with length < e there is a unique geodesic LEMMA 10.2. manifold M 7v: satisfying the conditions 7 (o) PROOF. = p, -- (0) dYv = v If we were willing to replace the interval (-2,2) by an arbitrarily small interval, then this statement would follow immediately from 10.1. To be more precise; there exists a neighborhood U of p0 and §10. numbers 11vII < 51 E1,e2 > 0 GEODESICS AND COMPLETENESS 57 for each p E U and each v E TMp with so that: there is a unique geodesic (-252,252) - M yv: satisfying the required initial conditions. To obtain the sharper statement it is only necessary to observe that the differential equation for geodesics has the following homogeneity property. Let c If the parametrized curve be any constant. t -F y(t) is a geodesic, then the parametrized curve t - y(ct) will also be a geodesic. Now suppose that Iti < 2 is smaller than e 5152. Then if ilvil < e and note that and IIv/5211 < 51 Hence we can define yv(t) to be le2tl <252 yv/e2(e2t) . This proves 10.2. . This following notation will be convenient. Let V E TMq be a tangent vector, and suppose that there exists a geodesic y: M [0,11 satisfying the conditions y(o) Then the point exponential = q, -(o) y(1) E M will be denoted by of the tangent vector v. = v. expq(v) The geodesic and called the y can thus be des- cribed by the formula y(t) = expq(ty) . The historical motivation for this terminology is the following. If M is the group of 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 matrices. The function expl: TMI -. M as defined above is then given by the exponential power series expl(A) = I + A + 2 A2 + A3 + ... 1 RIEM&NNIAN GEOMETRY II. 58 Lemma 10.2 says that is not defined for large vectors expq(v) In general, defined at all, expq(v) However, if v. is always uniquely defined. The manifold M is geodesically complete DEFINITION. is defined for all is defined providing that IIvjj is small enough. expq(v) q E M and all vectors v E TMq. expq(v) if This is clearly equiva- lent to the following requirement: For every geodesic segment to extend [a,b] yo: to an infinite geodesic 70 y: R-M We will return to a study of completeness . after proving some local results. Let TM be the tangent manifold of (p,v) if with p E M, (u1,...,un) ai consisting of all pairs the following TM q E U can be expressed uniquely as I au We give v E TMp. M, COO structure: is a coordinate system in an open set U C M then every tangent vector at where - M it should be possible t1 a1 +...+ tnan, Then the functions uconstitute q TU C TM. a coordinate system on the open set Lemma 10.2 says that for each p E M the map (q,v) - expq(v) is defined throughout a neighborhood V of the point more this map is differentiable throughout V. Now consider the smooth function F: F(q,v) (p,o) = (q, expq(v)). is non-singular. Further- (p,o) E TM. V - M x M defined by We claim that the Jacobian of F at the point In fact, denoting the induced coordinates on U x U C M x M by F( u F* T a Thus the Jacobian matrix of F u 2), we have a a +u2 _ a a = au? at (p,o) has the form ( o I 1 , and hence is non-singular. It follows from the implicit function theorem that F neighborhood V' of maps some (p,o) E TM diffeomorphically onto some neighborhood §10. of p such that (q,v) and such that 59 We may assume that the first neighborhood (p,p) e M X M. of all pairs GEODESICS AND COMPLETENESS belongs to a given neighborhood q Choose a smaller neighborhood W livII < s. consists v, of p U' of so that Then we have proven the following. F(V') D W x W. For each p e M there exists a neighborhood W and a number e > 0 so that: Any two points of W are joined by a unique (1) geodesic in M of length < e. LEMMA 10.3. (2) points. This geodesic depends smoothly upon the two 0 < t < 1, is the t -r exp q1(tv), (I.e., if then the pair (q1,v) e geodesic joining q1 and q2, TM depends differentiably on (g1,g2).) For each q e W the map expq maps the open (3) e-ball in TMq diffeomorphically onto an open set Uq) W. REMARK. With more care it would be possible to choose W so that the geodesic joining any two of its points lies completely within W. ptre J. H. C. Whitehead, Convex regions in the geometry of paths, Com- Quarter- ly Journal of Mathematics (Oxford) Vol. 3, (1932), pp. 33-42. Now let us study the relationship between geodesics and arc-length. THEOREM 10.4. Let W and e be as in Lemma 10.3. Let [0,1]- M y: be the geodesic of length < e joining two points of W, and let w: [0,1]- M be any other piecewise smooth path joining the same two points. Then, 1 1 SlI IIdt< 0 SII Udt 0 where equality can hold only if the point set w([o,1]) coincides with y([0,1]). Thus y is the shortest path joining its end points. The proof will be based on two lemmas. be as in 10.3. Let q = y(o) and let Uq II. RIEMANNIAN GEOMETRY 6o In Uq, LEMMA 10.5. the geodesics through q are the orthogonal trajectories of hypersurfaces expq(v) Let PROOF: v e TMq, : constant} = IIvii denote any curve in TMq with t -+ v(t) 11v(t)II 1. We must show that the corresponding curves t -+ expq(rov(t) ) in Uq, are orthogonal to the radial geodesics 0 < ro < e, where r -+ expq(rv(t0)) In terms of the parametrized surface f(r,t) f given by expq(rv(t)), = . 0 < r < e , we must prove that <T,> =o -5 E for all (r,t). Now a Fr of D of D of of of of Nr,a't> _ < cTr cTr'3f> + <Tr'Tr ; E> . The first expression on the right is zero since the curves r -+ f(r,t) are geodesics. The second expression is equal to of D of > <i, -JE .> = 2 aE a since IIII = dent of But for r. IIv(t) II 1. = of of = 0 Therefore the quantityCTF<,> is indepen-H we have r = o f(o,t) = expq(0) = q hence M(0,t) Therefore < Fr- 7E > is identically zero, which com- = 0. , pletes the proof. Now consider any piecewise smooth curve CO: Each point w(t) o < r(t) < e, [a,bl -+ Uq - (q) can be expressed uniquely in the form and IIv(t) II LEMMA 10.6. expq(r(t)v(t)) = 1, v(t) e TMq. The length equal to Ir(b) - r(a)I, function r(t) a IIdII dt is greater than or where equality holds only if the is monotone, and the function v(t) is constant. with §10. GEODESICS AND COMPLETENESS 61 Thus the shortest path joining two concentric spherical shells around is a radial geodesic. q PROOF: Let so that expq(rv(t)), f(r,t) Then m(t) = f(r(t),t) afr,(t) +af dm = CIE 3E cTr Since the two vectors on the right are mutually orthogonal, and since 1, this gives II Ir'(t)12 + = II2 where equality holds only if = b S a II2 > Ir'(t)12 II 0; hence only if dv = 0. Thus b -Ildt > S Ir'(t) Idt > Ir(b) - r(a) I a where equality holds only if v(t) is constant. is monotone and r(t) This completes the proof. The proof of Theorem 10.4 is now straightforward. from q piecewise smooth path m q' where 0 < r < e, IvIi to a point expq(rv) E Uq = 1. = Then for any s > o tain a segment joining the spherical shell of radius shell of radius segment will be > r - s; will be > r. If the path m must cons and lying between these two shells. r, hence letting m([0,1]) Consider any s tend to does not coincide with easily obtain a strict inequality. 0 to the spherical The length of this the length of y([0,1)), m then we This completes the proof of 10.4. An important consequence of Theorem 1o.4 is the following. Suppose that a path co: [o,A] - M, parametrized by arc-length, has length less than or equal to COROLLARY 10.7. the length of any other path from m(0) to is a geodesic. PROOF: Consider any segment of above, and having length 10.4. < e. Hence the entire path m DEFINITION. A geodesic co m(A). Then m lying within an open set W, as This segment must be a geodesic by Theorem is a geodesic. y: [a,b] -r M will be called minimal if 62 II. RIEMANNIAN GEOMETRY its length is less than or equal to the length of any other piecewise smooth path joining its endpoints. Theorem 10.4 asserts that any sufficiently small segment of a On the other hand a long geodesic may not be minimal. geodesic is minimal. For example we will see shortly that a great circle arc on the unit sphere If such an arc has length greater than is a geodesic. n, it is certainly not minimal. In general, minimal geodesics are not unique. For example two anti- podal points on a unit sphere are joined by infinitely many minimal geodesics. However, the following assertion is true. Define the distance p(p,q) p,q e M between two points to be the greatest lower bound for the arc-lengths of piecewise smooth paths joining these points. This clearly makes M into a metric space. It follows easily from 10.4 that this metric is compatible with the usual topology of M. Given a compact so that any two tance less than s are joined by length less than s. Furthermore and depends differentiably on its COROLLARY 10.8. a number PROOF. 6 > 0 Cover K by open sets set K C M there exists points of K with disa unique geodesic of this geodesic is minimal; endpoints. Wa, as in 10.3, and let small enough so that any two points in K with distance less than in a common Wa. be s s lie This completes the proof. Recall that the manifold M is geodesically complete if every geodesic segment can be extended indifinitely. THEOREM 10.9 (Hopf and Rinow*). If M is geodesically complete, then any two points can be joined by a minimal geodesic. PROOF. Up Given p,q e M with distance as in Lemma 10.3. Let Compare p. 341 of S C Up r > 0, choose a neighborhood denote a spherical shell of radius s < a G. de Rham, Sur la r6ductibilite d'un espace de Riemann, Commentarii Math. Helvetici, Vol. 26 (1952); as well as H. Hopf and W. Rinow, Ueber den Begriff der-vollstandigen differentialgeometrischen Flache, Commentarii,Vol. 3 (1931), pp. 209-225. GEODESICS AND COMPLETENESS §10. about is compact, there exists a point S Since p. p0 on S 63 expp(sv) = for which the distance to , 1, = 11V II We will prove that is minimized. q expp(rv) = This implies that the geodesic segment t is actually a minimal geodesic from p to q. 0 < t < r, y(t) = expp(tv), -+ q. The proof will amount to showing that a point which moves along the geodesic must get closer and closer y to In fact for each q. t e [s,r] we will prove that (it) p(y(t),q) This identity, for r-t = . will complete the proof. t = r, First we will show that the equality path from p to must pass through q p(p,q) = Min se S p(p0,q) = r - s. Therefore is true. (it) Since every we have S, (p(p,s) + p(s,q)) = s + p(po,q) Since p0 = y(s), this proves (1s). denote the supremum of those numbers t0 e [6,r] Let is true. (18) Then by continuity the equality t for which is true also. (it ) 0 If to < r we will obtain a contradiction. cal shell of radius point of S' about the point s' with minimum distance from p(Y(t0),q) = Let y(to); q. S' denote a small spheri- and let pp e S' (Compare Diagram 10.) be a Then p(s,q)) = 6' + p(pp,q) hence p(po,q) = (r - t0) - 6' (2) We claim that po is equal to y(to + s'). In fact the triangle inequality states that p(p,pp') > p(p,q) - p(pp,q) = to + s' (making use of (2)). po But a path of length precisely is obtained by following a minimal geodesic from y(t0) y from p to po. to y(to), to + s' from p to and then following Since this broken geodesic has minimal length, it follows from Corollary 10.7 that it is an (unbroken) II. 64 REIMANNIAN GEOMETRY geodesic, and hence coincides with Thus (1 y(to + s') = pp. 0(y(to + to+s') y. Now the equality (2) becomes ,'),q) = r - (to + s') This contradicts the definition of t0; and completes the proof. Diagram 10. As a consequence one has the following. If M is geodesically complete then every bounded subset of M has compact closure. Consequently M is complete as a metric space (i.e., every Cauchy sequence converges). COROLLARY 10.10. PROOF. expp: of If X C M has diameter TMp -+ M maps the disk of radius d d then for any p E X in TMp M which (making use of Theorem 10.9) contains of X the map onto a compact subset X. Hence the closure is compact. Conversely, if M is complete as a metric space, then it is not difficult, using Lemma 10.3, to prove that M is geodesically complete. For details the reader is referred to Hopf and Rinow. Henceforth we will not distinguish between geodesic completeness and metric completeness, but will refer simply to a complete Riemannian manifold. GEODESICS AND COMPLETENESS §10. 65 In Euclidean n-space, FAMILIAR EXAMPLES OF GEODESICS. with Rn, the usual coordinate system x1,...,xn and the usual Riemannian metric dx®® dx1 +...+ dxn ® dxn we have desic y, ri and the equations for a geo- 0 = given by t- (x1(t),...,xn(t) become d2xi dt whose solutions are the straight lines. follows: ' This could also have been seen as it is easy to show that the formula for arc length S i=1 ( \ 2dt coincides with the usual definition of aru 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 intersections of PROOF. Reflection through a plane with a unique geodesic C = Sn n E2. and E2 I(C') I(y) = y. Sn. is an isometry I: Sn -+ Sn Let x and y be two points of of minimal length between them. C' is an isometry, the curve between are precisely the great circles, that is, the Sn with the planes through the center of whose fixed point set is I(x) = x Sn Then, since is a geodesic of the same length as Therefore C' = I(C'). C I C' This implies that C' C C. Finally, since there is a great circle through any point of Sn in any given direction, these are all the geodesics. Antipodal points on the sphere have a continium of geodesics of minimal length between them. All other pairs of points have a unique geo- desic of minimal length between them, but an infinite family of non-minimal geodesics, depending on how many times the geodesic goes around the sphere and in which direction it starts. By the same reasoning every meridian line on a surface of revolution is a geodesic. The geodesics on a right circular cylinder Z are the generating lines, the circles cut by planes perpendicular to the generating lines, and 66 II. the helices on Z. PROOF: isometry I: RIEMANNIAN GEOMETRY L If is a generating line of Z - L - R2 by rolling Z Z then we can set up an onto R2: a1 The geodesics on Z are just the images under in R.2 Two points on I-1 of the straight lines Z have infinitely many geodesics between them. PART III THE CALCULUS OF VARIATIONS APPLIED TO GEODESICS §11. The Path Space of a Smooth Manifold. p and Let M be a smooth manifold and let sarily distinct) points of will be meant a map 1) [0,1] By a piecewise smooth path from p to q M. [0,1] - M such that w: there exists a subdivision 0 = to < t1 < ... < tk is differentiable of class so that each wl[ti_1,ti] 2) be two (not neces- q = 1 of C w(o) = p and w(1) = q. The set of all piecewise smooth paths from p by 0(M;p,q), or briefly by 0(M) Later (in §16) 0 or in M will be denoted q to 11. will be given the structure of a topological We will think of space, but for the moment this will not be necessary. To start the "infinite dimensional manifold." as being something like an a analogy we make the following definition. By the tangent space of a at a path w will be meant the vector space consisting of all piecewise smooth vector fields W along w which W(0) = 0 and W(1) = 0. for The notation TOW will be used for this vector space. If F is a real valued function on F*: it is natural to ask what 0 TQw - TRF(w) , the induced map on the tangent space, should mean. When F which is smooth in the usual sense, on a smooth manifold F*: TMp - T RKp) u - a(u) in M, as follows. Given X E TMp which is defined for a(0) = p, = M, we can define choose a smooth path -e < u < e moo) is a function X , so that 68 CALCULUS OF VARIATIONS III. Then F*(X) d(F(a(u))I is equal to Tu multiplied by the basis vector u=o' d Uf )F (P) E TRF(p) In order to carry out an analogqus construction for F: a R, - the following concept is needed. m A variation of DEFINITION. (keeping endpoints fixed) is a function (-e,e) -. 0, a: e > o, for some of such that m 1) &(o) 2) there is a subdivision = 0 = to < t1 < ... < tk = 1 so that the map [0,11 (-e,e) x [0,11 -+ M a: defined by is a(u,t) = &(u)(t) C°° on each strip (-e,e) x [ti-1,ti Q = 0(M;p,q), note that: i = 1,...,k. Since each 3) belongs to &(u) a(u,o) = p, a(u,1) = q We will use either a for all u e (-e,e) a or to refer to the variation. generally if, in the above definition, hood U of tion of in Rn, then a (or (-e,e) More is replaced by a neighbor- is called an n-parameter varia- a) m. Now vector" 0 . a cTu (o) d. may be considered as a "smooth path" in e Tom Its "velocity 11. is defined to be the vector field W along m given by Wt Clearly W E T11m. = m(o)t =(o,t) This vector field W is also called the variation vec- tor field associated with the variation a. Given any W E Tom note that there exists a variation (-e,e) - 11 which satisfies the conditions a(o) = to, mo(o) = W. In fact one can set &(u) (t) = expm(t)(u Wt) . By analogy with the definition given above, if F is a real valued function on we attempt to define a, F* : as follows. Given W e Tow d F(n)' = mo(o) w, d(F())lu=o equal to F*(W) -+ TRF(m) TO choose a variation a(o) and set 69 THE PATH SPACE §11. CLU = a: (-E,e) - 0 with W multiplied by the tangent vector Of course without hypothesis on F there is no guarantee that this derivative will exist, or will be independent of the choice of a. We will not investigate what conditions F must satisfy in order for F* We have indicated how F* to have these properties. might be defined only to motivate the following. DEFINITION. F: A path w 0 -+ R if and only if dF( u is a critical path for a function is zero for every variation I a of u=o w. EXAMPLE. derivatives ( If F takes on its minimum at a path mo, are all defined, then clearly mo and if the is a critical path. CALCULUS OF VARIATIONS III. 70 The Energy of a Path. §12. The length of a vec- Suppose now that M is a Riemannian manifold. i v E TMp will be denoted by for energy of from a to co o < a < b < 1) (where b For < v,v >2. _ IlvOO define the a e a as =SI1-df Ea(m) a We will write E for E. This can be compared with the arc-length from a to given by b b b La ces) = S IIdt II a Applying Schwarz's inequality as follows. b b b S fgdt)l2 < \ S f 2dt') ( .S g2dt)l a a with f(t) = 1 g(t) = and a we see that IIdo) II < (b - a)Ea (Lb where equality holds if and only if the parameter t , is constant; that is if and only if g is proportional to arc-length. Now suppose that there exists a minimal geodesic to q = m(1). y from p = m(o) Then E(y) = L ( - / ) L ( w ) . L(m)2 can hold only if m geodesic, possibly reparametrized. (Compare §10.7.) On the other hand Here the equality L(y)2 the equality L(w)2 to arc-length along a minimal geodesic. = w. = E(m) can hold only if the parameter is proportional This proves that E(y) < E(m) unless In other words: Let M be a complete Riemannian manifold p,q e M have distance d. Then the energy LEMMA. 12.1. and let is also a minimal function E: a(M;p,q) - R a is also TEE ENERGY OF A PATH §12. takes on its minimum geodesics from p precisely on the set of minimal d2 to q. Let are critical paths for the m e 9 We will now see which paths energy function E. (-e,e) - 2 a: be a variation of = = and let Wt = m, Tu- (o,t) Furthermore, let: be the associated variation vector field. Vt 71 m velocity vector of UE DcT At otv - = Vt. - Vt_ = = discontinuity in the velocity vector at 0<t<1 where otV Of course = 0 m acceleration vector of . for all but a finite number of values of THEOREM 12.2 (First variation formula). dE(a(u)) is equal to u=o 2 - t, t. The derivative 1 1 PROOF: - S <Wt,At> dt <Wt,AtV> t 0 According to Lemma 8.3, we have as as D as as Tu <-cf,ZT > = 2 < Tu Z' Therefore dE(a(u)) _ d TU- J( as, as > dt <Z 0 0 D as By Lemma 8.7 we can substitute Choose each strip (-e,e) x [ti_1,ti1. as as -ate D as 3u 3T in this last formula. for 3ETu 0 = t0 < t1 <...< tk = [ti_1,til, as follows. < D -3 as,- as > dt 2) a so that 1 Then we can is differentiable on "integrate by parts" on The identity D as ,as <uT as D as <au, >+ -3y > implies that ti S 3E 3u ,3T > dt < aa, aa Tu3t> = ti-1 (ti ,1 )a <Tu, D 3E>dt , 2T) ti-1 CALCULUS OF VARIATIONS III. 72 i = 1,...,k; Adding up the corresponding formulas for that u = o for t = 0 or 1, this gives k-1 1 2 Setting as pti < U, L i=1 dE(a(u)) u u = 0, and using the fact as 1 ]- as D as < y.-, ZE ZE > dt 0 we now obtain the required formula 2 dE a (o) < W,AtV > - S < W,A > dt t 0 This completes the proof. Intuitively, the first term in the expression for da(0) shows that varying the path w in the direction of decreasing "kink," tends to E; see Diagram 11. decrease The second term shows that varying the curve in the direction of its () tends to reduce acceleration vector Recall that the path w e 0 w of is w C`° is called a geodesic on the whole interval [0,11, is identically zero along COROLLARY 12.3. function E E. if and only if and the acceleration vector -( w. The path w if and only if co is a critical point for the is a geodesic. THE ENERGY OF A PATH §12. PROOF: Clearly a geodesic is a critical point. There is a variation of critical point. f(t) 73 m with W(t) = f(t)A(t) is positive except that it vanishes at the 2 mo(o) _ m be a Let ti. where Then - f(t) < A(t),A(t) > dt. 0 This is zero if and only if A(t) = 0 for all t. Hence each nI[ti,ti+1] is a geodesic. Now pick a variation such that W(ti) = Ati V. 2 (0) _ m < AtiV,AtiV > . is differentiable of class Then If this is zero then all AtV C1, even at the points ti. from the uniqueness theorem for differential equations that everywhere: thus co is an unbroken geodesic. are 0, and Now it follows m is C°° CALCULUS OF VARIATIONS III. 74 The Hessian of the Energy Function at a Critical Path. §13. Continuing with the analogy developed in the preceding section, we now wish to define a bilinear functional E**: when -R TOY x TOy is a critical point of the function y i.e., a geodesic. E, bilinear functional will be called the Hessian of If point p, f E at This y. is a real valued function on a manifold M with critical then the Hessian f**: can be defined as follows. (u1,u2) a(u1,u2) values in M, TM p x TMp -+ R Given X1,X2 E TMp choose a smooth map defined on a neighborhood of (0,0) in R2, with so that a(o,o) p, = = X1(0,0) X2(0,0) x1 = x2 Then 2f(a(u1,u2)) f**(X1,X2) = au1 au2 This suggests defining E** as follows. (0,0) Given vector fields W1,W2 e Tsar choose a 2-parameter variation a: where U is a neighborhood of a(o,o,t) = y(t), Ux[0,11 -M (0,0) in R2, (o,o,t) so that W1(t), = au (0,0,t) (Compare §11.) = W2(t) 2 1 Then the Hessian E**(W1,W2) will be defined to be the second partial derivative 2E(a(ut,u2))I auI au2 where a(u1,u2) e 0 denotes the path (0 , 0) a(u1,u2)(t) = a(u1,u2,t) . This second derivative will be written briefly as o u2(0,0) 1 2 The following theorem is needed to prove that E** is well defined. THE HESSIAN OF E §13. 75 THEOREM 13.1 (Second variation formula). Let be a 2-parameter variation of the geodesic variation vector fields Wi = x,(0,0) e Tay, 1 Then the second derivative 2 &: U -+sa with y i = 1,2 a"E of the energy (0,0) u3 you function is equal to DW - D 2W <W2(t),AtUU1> -S t I < W21 + R(V,W1)V > dt 0 where V = d denotes the velocity vector field and where ofd _(t+) -(t ) DW denotes the jump in -dy at one of its finitely many points of discontinuity in the open unit interval. PROOF: According to 12.2 we have 1 1 aE 2 3u2 = - as as as D as At < cTu2,t > - S <Tu2, WE 3T > t 0 Therefore 1 f a2E u2 u1 = - D t as > - 3T as D C ac < cTu1 D as D as 2P1 t as > dt < cTu1 Tu2 ) 3T 'R as At 'R > D D as 37 0 0 Let us evaluate this expression for (u1,u2) = (0,0). Since y = &(0,0) is an unbroken geodesic, we have as At 3T = 0, D as 0 WE ZT e so that the first and third terms are zero. Rearranging the second term, we obtain (13.2) 1 2 6E 0 0) - W2 At aE W1 < W2, 1 -RV > dt ) 0 In order to interchange the two operators and D 3 , we need to 1 bring in the curvature formula, 1 D V - A 1V = R( 4,Tu- )V 1 = R(V,W1)V dt. 76 CALCULUS OF VARIATIONS III. V Together with the identity = 1 (13.3) D _3T = 3U T_ 1 _CTU this yields W1 1 DST V + R(V,W1)V = d ff Substituting this expression into (13.2) this completes the proof of 13.1. 2 )E The expression E**(W1,W2) = COROLLARY 13.1+. -2(0,0) is a well defined symmetric and bilinear function of W and W2. PROOF: The second variation formula shows that depends only on the variation vector fields This formula also shows that is well defined. E**(W1,W2) and W2, W1 ua u 7-0, 0) 2 1 so that E** is bilinear. The symmetry property E**(W1,W2) E**(W2, W1) = is not at all obvious from the second variation formula; but does follow immediately from the symmetry property REMARK 13.5. E** = aEu2 u1 The diagonal terms uc E**(W,W) 6 E 7 1 of the bilinear pairing can be described in terms of a 1-parameter variation of E**(W,W) where (-e,e) -+s1 a: In fact a(o) denotes any variation of equal to W. field T- (o) d E = y. with variation vector y To prove this identity it is only necessary to introduce the two parameter variation (u1,u2) = a(u1 + u2) and to note that u=ci' t a9 da a2E ° u = d2 d2E ° a As an application of this remark, we have the following. LEMMA 13.6. If y is a minimal geodesic from p to q then the bilinear pairing E** is positive semi-definite. Hence the index X of E** is zero. PROOF: d2E(a( , The inequality evaluated at E(a(u)) > E(y) u = 0, is > 0. Hence = E(a(o)) implies that E**(W,W) > 0 for all W. JACOBI FIELDS §14. The Null Space of Jacobi Fields: §14. A vector field along a geodesic J 77 E** is called a Jacobi field y if it satisfies the Jacobi differential equation 2 D + R(V,J)V = o dt where V = This is a linear, second order differential equation. Ur . [It can be put in a more familiar form by choosing orthonormal parallel vector fields along P1,...,Pn Then setting y. J(t) = E fi(t)Pi(t), the equation becomes 2 i da + L a'(t)fj(t) o, = i = 1,..., n; j=1 where a3 = < R(V,Pj)V,Pi > .] Thus the Jacobi equation has 2n linearly independent solutions, each of which can be defined throughout solutions are all A C°°-differentiable. given Jacobi field J y. The is com- pletely determined by its initial conditions: J(o), uE(o) e TM 'Y(0) Let with a and p = y(a) q = y(b) be two points on the geodesic y, b. and p DEFINITION. J non-zero Jacobi field p The multiplicity of q along and q are conjugate* along y y which vanishes for if there exists a t = a and t = b. as conjugate points is equal to the dimen- sion of the vector space consisting of all such Jacobi fields. Now let y be a geodesic in S2 = S0(M;p,q). Recall that the null- space of the Hessian E**: TOY x TOy--. R is the vector space consisting of those * W1 e TOy such that E**(W1,W2) = 0 If y 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. CALCULUS OF VARIATIONS III. 78 for all W2. The nullity E** null space. v is degenerate is equal to the dimension of this E** of if v > 0. THEOREM 14.1. A vector field W1 e TOy belongs to the null space of E** if and only if W1 is a Jacobi field. Hence E** is degenerate if and only if the end points p and q are conjugate along y. The nullity of E** is equal to the multiplicity of p and q as conjugate points. (Compare the proof of 12.3.) PROOF: p vanishes at and then J q, J. is a Jacobi field which If certainly belongs to The second TOO. variation formula (§13.1) states that Cl t Hence 0 belongs to the null space. J Conversely, suppose that W1 Choose a subdivision W1l[ti_1,ti1 belongs to the null space o = t0 < t1 <...< tk = is smooth for each Let i. of 1 E**. so that [0,11 [0,11 f: function which vanishes for the parameter values of be a smooth -+ [0,11 t0,t1,...,tk and is positive otherwise; and let D 2W 2 + R(VW1)V)t W2(t) = f(t)( dt Then 2E**(W1,W2) f(t)t1 0 + = 2 2W 1 - E o Since this is zero, it follows that W1j[ti-1,ti1 each dt + R(V,W1)VII is a Jacobi field for i. DW1 I Now let W2 e TOOy be a field such that W2(ti) = At i i = 1,2,...,k-1. -- for Then k-1 DW1 Ati - 2E**(W1,WI) 2 .- 1 0 dt = 0 + 0 i=1 DW1 Hence -cTE has no jumps. But a solution W1 of the Jacobi equation is DW1 completely determined b by the vectors W 1(ti) lows that the k Jacobi fields W11[ti_1,ti1, to give a Jacobi field W1 which is and Ur-(ti). i = 1,...,k, Thus it folfit together C`°-differentiable throughout the JACOBI FIELDS §14. 79 This completes the proof of 14.1. entire unit interval. It follows that the nullity is always finite. E** of v For there are only finitely many linearly independent Jacobi fields along Actually the nullity REMARK 14.2. satisfies v precisely it is clear that n, This will imply that V = 7 denotes the velocity vector field. DJ V+ t DV 1 (IT = but not for t = 0, In fact let v < n. has dimension We will construct one v < n. example of a Jacobi field which vanishes for t = 1. Since 0 < v < n. t = 0 the space of Jacobi fields which vanish for y. Jt = tVt where Then V 2 Uy since R = 0 hence = o), (Since dt7 = R(V,J)V = tR(V,V)V Thus is skew symmetric in the first two variables. satisfies the Jacobi equation. J Furthermore 0. J 0 = 0, Since J1 # 0, this completes the proof. Suppose that M is "flat" in the sense that the curva- EXAMPLE 1. ture tensor is identically zero. dt2 = 0. J(t) =) fi(t)Pi(t) Setting this becomes Then the Jacobi equation becomes 2 i ddf7 = 0. where Pi are parallel, Evidently a Jacobi field along y can have Thus there are no conjugate points, and E** at most one zero. is non-degenerate. EXAMPLE 2. unit sphere Sn, and let Then we will see that and p Suppose that be a great circle arc from p to y and p are antipodal points on the q q are conjugate with multiplicity Thus in this example the nullity n-1. largest possible value. q. v of E** takes its The proof will depend on the following discussion. Let a be a 1-parameter variation of the endpoints fixed, such that each x [0,11 a: be a by COO map such that &(u)(t) = a(u,t)] a(o,t) &(u) = y(t), is a geodesic. y, not necessarily keeping is a geodesic. That is, let -+ M and such that each a(u) [given 80 LEMMA 14.3. III. CALCULUS OF VARIATIONS a is such a variation of If y through geodesics, then the variation vector field W(t) _(O,t) is a Jacobi field along a is a variation of If PROOF: is identically zero. 0 ° through geodesics, then y D as dT3£ Hence D as D D D as Tu3-E3y - 3TTu 3T D2 (Compare §13.3.) Y. as as R( as as \ as + ZE CTU a(X ) as + R\.t'Tu at Therefore the variation vector field Y is a Jacobi field. Thus one way of obtaining Jacobi fields is to move geodesics around. Now let us return to the example of two antipodal points on a unit n-sphere. Rotating the sphere, keeping p vector field along the geodesic and q. and q fixed, the variation will be a Jacobi field vanishing at p y Rotating in n-1 different directions one obtains n-1 linearly independent Jacobi fields. multiplicity n-1. Thus p and q are conjugate along y with §14. JACOBI FIELDS 81 Every Jacobi field along a geodesic y: [0,11-+ M through geodesics. y LEMMA 14.4. may be obtained by a variation of Choose a neighborhood U PROOF: of of so that any two points y(o) U are joined by a unique minimal geodesic which depends differentiably on the endpoints. Suppose that construct a Jacobi field W along values at t = 0 and Similarly choose b: (-E,E) We will first 0 < t < s. with arbitrarily prescribed yl[o,s] Choose a curve t = s. and so that mo(o) a(o) = y(o) for y(t) e U (-e,E) a: is any prescribed vector in -+ U with b(0) y(s) so that -+ U TM y(0). and T(o) arbitrary. Now define the variation a: (-E,E) x [o,s] by letting to b(u). -+ M M be the unique minimal geodesic from a(u): Then the formula t -+(o,t) (TU a(u) defines a Jacobi field with the given end conditions. Any Jacobi field along 7(y) yl[o,s] can be obtained in this way: denotes the vector space of all Jacobi fields W along formula W (W(0), W(s)) dimension 2n then the defines a linear map f: 7(y) We have just shown that f y, If -+ TM is determined by its values at f TM 7(s) Since both vector spaces have the same is onto. it follows that Y(0) X is an isomorphism. y(o) and y(s). I.e., a Jacobi field (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 yl[o,s1. The restriction of If u a(u) to the interval 10,81 is not essential. is sufficiently small then, using the compactness of [0,11, can be extended to a geodesic defined over the entire unit interval a(u) [0,11. This yields a variation through geodesics: a&: (-E',E') X [0,11 -+ M with any given Jacobi field as variation vector. REMARK 14.5. This argument shows that in any such neighborhood the Jacobi fields along a geodesic segment in U are uniquely determined U 82 III. CALCULUS OF VARIATIONS by their values at the endpoints of the geodesic. REMARK 14.6. (-S,s) of The proof shows also, that there is a neighborhood so that if 0 t e (-s,s) y(t) is not conjugate to We will see in §15.2 that the set of conjugate points to y(o) along y(o) along the entire geodesic y. then y has no cluster points. The index §15. THE INDEX THEOREM §15. The Index Theorem. 83 of the Hessian % E**: TOy x TOy-+ R is defined to be the maximum dimension of a subspace of Tay on which E** We will prove the following. is negative definite. THEOREM 15.1 (Morse). The index X of E** is equal y(t), with 0 < t < 1, such that y(t) is conjugate to y(o) along y; each such conjugate point being counted with its multiplicity. This index is always finite*. to the number of points . As an immediate consequence one has: COROLLARY 15.2. A geodesic segment y: [0,11 - M can contain only finitely many points which are conjugate to y(o) along y. In order to prove 15.1 we will first make an estimate for splitting the vector space one of which E** into two mutually orthogonal subspaces, on is contained in an open set y(t) o = to < t1 ..< tk = so that each segment that each Let such that any two y[ti_i,til yl[ti_1,tiI Choose a subdivision of the unit interval which is sufficiently fine 1 lies within such an open set TOy(to,tl,t2,...,tk) C TOy y and so be the vector space consisting of such that is a Jacobi field along 1) WI[ti_i,til 2) W vanishes at the endpoints T11y(t0,tl,...,tk) U; is minimal. all vector fields W along yl[ti_1,tiI for each i; t = 0, t = 1. is a finite dimensional vector space consisting of broken Jacobi fields along * U U are joined by a unique minimal geodesic which depends differ- entiably on the endpoints. (See §10.) Thus by is positive definite. Each point points of Tay ) y. For generalization of this result see: W. Ambrose, The index theorem in Riemannian geometry, Annals of Mathematics, Vol. 73 (1961), pp. 49-86. CALCULUS OF VARIATIONS III. 84 Let be the vector space consisting of all vector fields T' C Tsar for which W(t0) = 0, W E TOY W(t1) = 0, W(t2) = 0,..., W(tk) = o. LEMMA 15.3. The vector space Tsar splits as the direct sum TOy(to,t,,...,tk) ® T'. These two subspaces are mutually perpendicular with respect to the inner product restricted to E** Furthermore, E**. is positive T' definite. Given any vector field W E TOY PROOF: "broken Jacobi field" in TOy(t0,t1,...,tk) It follows from §14.5 that i = 0,1,...,k. Clearly W - W1 and belongs to generate T' If denote the unique W1 exists and is unique. W1 Thus the two subspaces, T'. for Tsay(to,t1,...,tk) and have only the zero vector field in common. Tsar, belongs to W1 let such that W1(ti) = W(ti) and W2 Tsay(to)tl,...,tk) belongs to T', then the second variation formula (13.1) takes the form <W2(t),pt -liE**(Wj 1 LW -1 <W2,o > dt > - t 0 = 0 Thus the two subspaces are mutually perpendicular with respect to E** For any W E TO y can be interpreted as the the Hessian E**(W,W) 2 d E 2 a (0); second derivative y where 7, with variation vector field mo(o) W belongs to points for y(ti) = Proof that & for W e T'. y(o) to y(t1) E(a(u)) > E(y) to Proof that E**(W,W) > 0 were equal to a(u) e 0 to ... to is a piece- y(1). But = This proves that E(a(o)) 0. for W E T', . u = 0, W 0. must be > 0. Suppose that Then W would lie in the null space of In fact for any W1 e T0y(t0,t1,...,tk) 0. Each y(t2) Therefore the second derivative, evaluated at = is chosen so as to leave the is a minimal geodesic, and therefore has smaller energy yl[ti_1,ti] E**(W1,W) If In other words we may assume that than any other path between its endpoints. E**(W,W) (Compare 13.5.) W. i = 0,1,...,k. E**(W,W) > 0 wise smooth path from each fixed. is any variation of (-s,e) equal to then we may assume that T' y(to),y(t1),...,y(tk) a(u)(ti) &: For any W2 e T' we have already seen that the inequality E**. THE INDEX THEOREM §15. 0 < E**(W + c W2, W + c W2) for all values of null space. T' implies that c But the null space of 85 2c E**(W2,W) + C = E**(W2,W) = 0. E** 2 E**(W2,W2) Thus W lies in the consists of Jacobi fields. Since contains no Jacobi fields other than zero, this implies that W = 0. Thus the quadratic form E** is positive definite on This T'. completes the proof of 15.3. An immediate consequence is the following: LEMMA 15.4. The index (or the nullity) of E** is equal to the index (or nullity) of E** restricted to the space TOy(t0,t1,...,tk) of broken Jacobi fields. In particular (since To (t0,tl,...,tk) is a finite dimensional vector 7 space) the index X is always finite. The proof is straightforward. Let Thus denote the restriction of yT [O,T] -+ M is a geodesic from yT: the index of the Hessian Thus X(1) ( Eo )** to the interval y y(o) to y(T). Let [O,T]. denote ?(T) which is associated with this geodesic. is the index which we are actually trying to compute. First note that: ASSERTION (1). X(T) For if then there exists a T < T' vector fields along Hessian ( Eo )** yT is a monotone function of which vanish at X(T) y(o) y(T) dimensional space and y(T) and y(T'). Thus we obtain a yT' X(T) P of such that the is negative definite on this vector space. field in '' extends to a vector field along between T. Each vector which vanishes identically dimensional vector space Eoi of fields along yT, on which )** is negative definite. Hence %(T) < ASSERTION (2). For if hence X(T) = 0 T T.(-T) = 0 for small values of is sufficiently small then yT T. is a minimal geodesic, by Lemma 13.6. Now let us examine the discontinuities of the function note that X(T) is continuous from the left: ASSERTION (3). A.(T-E) = X(T). X(T). For all sufficiently small e > 0 we have First 86 III. CALCULUS OF VARIATIONS x(1) can be interpreted as According to 15.3 the number PROOF. the index of a quadratic form on a finite dimensional vector space T0y(to,t1,...,tk). say We may assume that the subdivision is. chosen so that ti < T < ti+1. Then the index of a corresponding quadratic form broken Jacobi fields along using the subdivision can be interpreted as the index X(T) on a corresponding vector space of HT This vector space is to be constructed yT. 0 < t1 < t2 <... < ti < T of Since a [0,T1. broken Jacobi field is uniquely determined by its values at the break points y(ti), this vector space is isomorphic to the direct sum E = TMy(t1) ® TMy(t2) ® ... ® TMy(ti) ratic form H. Now X(T). on E HT Evidently the quad- is independent of T. Note that this vector space E varies continuously with . T. of dimension is negative definite on a subspace V C E sufficiently close to For all r' negative definite on V. it follows that T then we also have ?V(T-e) < >,(T) by Assertion 1. Hence ASSERTION (4). Let v e > 0 .l.(T+e) x(t) ( Eo ). we have x(T) + V = jumps by a conjugate point of multiplicity < T 'X(T-e) _ X(T). be the nullity of the Hessian Then for all sufficiently small Thus the function T' = T - e But if X(T') > Therefore is HT, when the variable v passes t and is continuous otherwise. v; Clearly this assertion will complete the proof of the index theorem. PROOF that of Assertion 3. a(T+e) < X(T) + v Since some subspace )' C Z ly close to T, HT Let we see that dim E = ni of dimension it follows that . fields along yT, X(T+e) > X(T) + v. is positive definite on For all T' sufficient- is positive definite on n". X(T') < dim E - dime" PROOF that H. ni - X(T) - v. HT, E be as in the proof and l.(T) + v Let W1,-..,wx(T) be X(T) vanishing at the endpoints, such that the matrix ( Ep )+ (Wi,Wj) Hence 1 vector THE INDEX THEOREM §15. is negative definite. fields along Let J1,...,J, be 87 linearly independent Jacobi v Note that the also vanishing at the endpoints. yT, v vectors DJh E TMY(T) -HE('r) are linearly independent. along X1, ... ,X, Hence it is possible to choose is equal to the v x v over DJh \\ dt Extend the vector fields Wi identity matrix. by setting these fields equal to yT+E vector fields YT+E' vanishing at the endpoints of rT + E, so that ( Jh v T < t < T + E. for 0 and Using the second variation formula we see easily that Toe )** Eo Now let Jh' WO = o Eo+EI**( Jh' Xk) = 2shk be a small number, and consider the c W1,...,Wx(T), along YT+E' c_1 J1 - c X1,..., c on which the quadratic form )(T) + v definite. In fact the matrix of vector fields Jv - c Xv is negative c A -4I+ c 2 c At and B ( E"'),, (Eo+E)** with respect to this basis is ET )**( WI,W A X(T) + v _1 We claim that these vector fields span a vector space of dimension where (Kronecker delta). are fixed matrices. If c B / is sufficiently small, this compound matrix is certainly negative definite. This proves Assertion (4). The index theorem 15.1 clearly follows from the Assertions (2),(3), and (4) . 88 CALCULUS OF VARIATIONS III. §16. A Finite Dimensional Approximation to 0c Let M be a connected Riemannian manifold and let two (not necessarily distinct) points of piecewise C°° paths from p to and a = 0(M;p,q) The set M. p can be topologized as follows. q be q of Let denote the topological metric on M coming from its Riemann metric. with arc-lengths w, W' e n s(t), 1 Max 0< t< 1 \ 2 .ds'\ dt] C r p(a(t), W'(t)) + / L Given respectively,define the distance s'(t) to be d(W,W') p J0 \ i z 1 (The last term is added on so that the energy function 2 Eb(W) = dt ( d a will be a continuous function from induces the required topology on Given and let c > 0 Int Sac 0c let to the real numbers.) This metric D. E-1([O,cl) C n denote the closed subset denote the open is the energy function). a subset E-1([o,c)) E = E1: n - R (where nc We will study the topology of by construct- ing a finite dimensional approximation to it. Choose some subdivision val. W: Let Sa(to,t1,...,tk) 0 = to< t1 <...< tk = be the subspace of a I of the unit inter- consisting of paths (0,11 - M such that 1) a(o) = p and a)(1) =q 2) ml[ti-1,ti) is a geodesic for each i = 1,...,k. Finally we define the subspaces n(to,t1,...,tk)c = nc n n(to,t1,...,tk) Int n(to,t1,...,tk)c = (Int 0c) n n(to,...,tk) Let M be a complete Riemannian manifold; and let c be a fixed positive number such that Dc / 0. Then for all sufficiently fine subdivisions (to,tl,...,tk) of [0,11 the set Int n(to,tl,...,tk)c can be given the LEMMA 1"6.1. structure of a smooth finite dimensional manifold in a natural way. AN APPROXIMATION TO §16. PROOF: Let 89 sac denote the ball S {x E M : o(x,p) < %r-c1 Note that every path w E sac lies within this subset from the inequality L2 < E < c . Since M S e > 0 so that whenever exists is complete, unique geodesic from x depends differentiably on x and Choose the subdivision difference x, y E S to y of length there there is a p(x,y) < s and so that this geodesic < e; y. (t0,t1,...,tk) is less than ti - ti-1 This follows Hence by 10.8 is a compact set. and S C M. e2/c. of [0,11 so that each Then for each broken geodesic w E o(t0,t1,...,tk)c we have ti ti 2 Lti-1 cu) = (ti - ti_1) ( Eti-1 cu) < (ti (E w) < (ti - ti_1)c < e2 mj[ti_1,ti1 Thus the geodesic is uniquely and differentiably determined by the two end points. The broken geodesic w is uniquely determined by the (k-1)-tuple 0)(t1), w(t2),...,w(tk_1) E M X M x...x M. Evidently this correspondence w --F (cu(t1),...,aa(tk-1)) defines a homeomorphism between Int 0(t0,t1,...,tk)c subset of the (k-1)-fold product M x...x M. and a certain open Taking over the differentiable structure from this product, this completes the proof of 16.1. To shorten the notation, let us denote this manifold Int 0(t0,t1,...,tk)c of broken geodesics by B. Let E': B -'R denote the restriction to B of the energy function E : 0 - it. 9o III. CALCULUS OF VARIATIONS R is smooth. This function E': B Furthermore, for each a < c the set Ba = (E')-1[o,a] is compact, and is a deformation retract* of the corresponding set 0a. The critical points of E' are precisely the same as the critical points of E in Int 0c: namely the unbroken geodesics from p to q THEOREM 16.2. of length less than '. The index or the nullity] of the Hessian E'** at each such critical point y E** at is equal to the index [ or the nullity ] of provides a faithful model Thus the finite dimensional manifold B for the infinite dimensional path space y. As an immediate conse- Int no. quence we have the following basic result. Let M be a complete Riemannian manifold p,q e M be two points which are not conjugate along any geodesic of length < %ra. Then as has the homotopy type of a finite CW-complex, with one cell of dimension X for each geodesic in 0a at which E** has index X. THEOREM 16.3. and let (In particular it is asserted that na contains only finitely many geodesics.) This follows from 16.2 together with PROOF. PROOF of 16.2. on the Since the broken geodesic §3.5. w e B depends smoothly (k-1)-tuple w(t1),w(t2),...,w(tk-1) E M x...x M it is clear that the energy El(w) tuple. also depends smoothly on this In fact we have the explicit formula k p(w(ti_1),a(ti))2/(ti _ ti_1) E'(w) i=1 Similarly B itself is a deformation retract of Int sac. (k-1)- the set Ba a < c For 91 is homeomorphic to the set of all (k-1)- (p1,...Pk_1) e S X S x...X S tuples 0c AN APPROXIMATION TO §16. such that k p(pi_1,pi)2/(ti - ti_1) < a i=1 po = p, (Here it is to be understood that pk = q.) As a closed subset of a compact set, this is certainly compact. A retraction Int 0c , B r: is defined as follows. from < e r(e) such that each r(w)I[ti_1,ti] denote the unique broken geodesic in B a geodesic of length Let to w(ti_1) w(ti). is The inequality p(p,w(t))2 < (L e)2 < E w < c implies that Hence the inequality w[o,1] C S. t. 2 p(w(t1_1),w(ti)) w) < < (ti - ti_1)( Et' . c = e2 i- implies that can be so defined. r(w) Clearly E(r(a)) < E(e) < c. This retraction r fits into a 1- parameter family of maps Int 0c -+ Int sa c ru : For as follows. t1-1 < U < ti let ru(e)I[o,ti_1I = r(e)I[o,ti-1] ru(w)I[ti_1 u] = minimal geodesic from , e(ti_1) to e(u) , ru(w)I[u,1] Then B . is the identity map of ro fied that that eI[u,1] = ru(w) r1 = r. It is easily veri- is continuous as a function of both variables. is a deformation retract of Since and Int 0c, E(ru(w)) < E(w) mation retract of This proves Int Sac. it is clear that each Ba is also a defor- saa. Every geodesic is also a broken geodesic, so it is clear that every "critical point" of E in Int sac automatically lies in the submanifold Using the first variation formula (§12.2) points of E' are precisely the unbroken geodesics. Consider the tangent space Y. it is clear that the critical TBy to the manifold B This will be identified with the space T0y(to)tt,...,tk) at a geodesic of broken B. III. 92 Jacobi fields along as described in §15. y, justified as follows. CALCULUS OF VARIATIONS This identification can be Let B be any variation of y variation vector field through broken geodesics. (o,t) along y Then the corresponding is clearly a broken Jacobi field. (Compare §14.3) Now the statement that the index (or the nullity) of is equal to the index (or nullity) of quence of Lemma 15.4. REMARK. E,* at y E** at is an immediate conse- This completes the proof of 16.2. As one consequence of this theorem we obtain an alternative proof of the existence of a minimal geodesic joining two given points of a complete manifold. ponding set function y E': y For if oa(p,q) is non-vacuous, then the corres- Ba will be compact and non-vacuous. Hence the continuous Ba _ R will take on its minimum at some point will be the required minimal geodesic. p,q y E Ba. This THE FULL PATH SPACE §17. §17. 93 The Topology of the Full Path Space. Let M be a Riemannian manifold with Riemann metric p be the induced topological metric. sarily distinct) points of Let p and q g, and let be two (not neces- M. In homotopy theory one studies the space 0* of all continuous paths w: from p to -M [0,11 in the compact open topology. q, This topology can also be described as that induced by the metric M x plw(t),w'(t) t On the other hand we have been studying the space paths from p to q 0 of piecewise with the metric i 1 dc°,w') Since > d* d = d*(°, ) + r L C°° S 0 (dT - ')2 dt I 2 J the natural map is continuous. THEOREM 17.1. lence between This natural map 0 and f*. [Added June 1968. i is a homotopy equiva- The following proof is based on suggestions by W. B. Houston, Jr., who has pointed out that my original proof of 17.1 was incorrect. S2* The original proof made use of an alleged homotopy inverse O which in fact was not even continuous.] PROOF: We will use the fact that every point of M has an open neighborhood N which is "geodesically convex" in the sense that any two points of N are joined by a unique minimal geodesic which lies completely within N and depends differentiably on the endpoints. to J. H. C. Whitehead. (This result is due See for example Bishop and Crittenden, "Geometry of 94 CALCULUS OF VARIATIONS III. manifolds," p. 246; Helgason, "Differential geometry and symmetric spaces," p. 53; or Hicks, "Notes on differential geometry," p. 134.) Choose a covering of M by such geodesically convex open sets Na Subdividing the interval [0,1] into 2k equal subintervals [(j-l)/2k,j/2k], let Qk denote the set of all continuous paths w from [(j-l)/2 k,j/2k] should be contained in one of the sets Clearly each C1 p to q, and clearly C2 p to q which sat- the image under m of each subinterval isfy the following condition: is an open subset of the space Na of the covering. S2* of all paths from is the union of the sequence of open subsets SLR C C2 C 0 C Similarly the corresponding sets "k = i-1(C1 ) are open subsets of 0 with union equal to 0. We will first show that the natural map "k -3 'k (iIC<) is a homotopy equivalence. For each w E OO let h(w) E S2k be the broken geodesic which coincides with w for the parameter values t = j/2k, j = 0,1,2,...,2k, and which is a minimal geodesic within each intermediate interval [(j-1)/2k, j/2k). This construction defines a function h : 0 -* 0k and it is not difficult to verify that h is continuous. Just as in the proof of 16.2 on page 91, it can be verified that the composition (ilCak) ° h composition h e (iI0k) is homotopic to the identity map of Ck and that the is homotopic to the identity map of cc. This proves that ilQlc is a homotopy equivalence. To conclude the proof of 17.1 we appeal to the Appendix. Using Ex- ample 1 on page 149 note that the space 0 is the homotopy direct limit of the sequence of subsets S. Similarly note that 4* is the homotopy direct limit of the sequence of subsets Sc. that i : 0-* S3* Therefore, Theorem A (page 150) shows is a homotopy equivalence. This completes the proof. THE FULL PATH SPACE §17. It is known that the space 95 has the homotopy type of a st* OW- (See Milnor, On spaces having the homotopy type of a CW-complex, complex. COROLLARY 17.2. 0 Therefore pp. 272-280.) Trans. Amer. Math. Soc., Vol. 9o (1959), has the homotopy type of a CW- complex. This statement can be sharpened as follows. THEOREM 17.3. (Fundamental theorem of Morse Theory.) Let M be a complete Riemannian manifold, and let p,q E M be two points which are not conjugate along Then 0(M;p,q) (or 0*(M;p,q)) has the any geodesic. homotopy type of a countable CW-complex which contains one cell of dimension to of index q for each geodesic from p % X. The proof is analogous to that of 3.5. a0 < a1 < a2 < ... energy function Choose a sequence of real numbers which are not critical values of the E, so that each interval one critical value. Consider the sequence 0ao C ,al C (a1_1,ai) ,a2 C ... contains precisely ; a where we may assume that with 3.3 and 3.7 that each 0 ai has the homotopy type of finite number of cells attached: in K0 C K1 C K2 C I K0 of homotopy equivalences. Since 0 stal f C C K1 Letting ;tae C ... C K2 C ... 0 - K be the direct limit mapping, f: induces isomorphisms of homotopy groups in all dimen- is known to have the homotopy type of a OW-complex (17.2) completes the proof. EXAMPLE. f is a homotopy equivalence. This [For a different proof, not using 17.2, see p. 149.1 The path space of the sphere are two non-conjugate points on p' % 1 1 it follows from Whitehead's theorem that where with a CW-complexes with cells of the required of ... ,a0 C sions. together one %-cell for each geodesic of index description, and a sequence it is clear that 9 ai-1 Now, just as in the proof of 3.5, one constructs a se- E-1(a1_1,ai). quence It follows from 16.2 is vacuous. o st Suppose that p and q That is, suppose that Sn. denotes the antipode of Sn. p. a / p,p' Then there are denumerably many CALCULUS OF VARIATIONS III. 96 geodesics from p yo'y1' 2' short great circle arc from p let pq'p'q; circle are to to q, as follows. let q; k denotes the number of times that subscript interior of denote the yo and so on. pgp'q'pq; denote the arc y2 Let denote the long great y1 p or The occurs in the p' yk. The index x(yk) = µ1 +...+ µk p plicity Therefore we have: n-1. p' since each k(n-1), in the interior is conjugate to of the points or is equal to p with multi- has the homotopy The loop space st(Sf) CW-complex with one cell each in the dimensions COROLLARY 17.4. type of a 0, n-1, 2(n-1), 3(n-1),... For n > 2 from this information. the homology of Since Q(Sn) can be computed immediately Q(Sn) has non-trivial homology in infinite- ly many dimensions, we can conclude: COROLLARY 17.5. for n > 2. M Let have the homotopy type of Then any two non-conjugate points of Sn, M are joined by infinitely many geodesics. This follows since the homotopy type of 0(M)) depends only on the homotopy type of geodesic in 3(n-1), 0(M) and so on. with index 0, M. o*(M) (and hence of There must be at least one at least one with index n-1, 2(n-1), §17. REMARK. THE FULL PATH SPACE More generally if M is any complete manifold which is not contractible then any two non-conjugate points of infinitely many geodesics. 97 M are joined by Compare p. 484 of J. P. Serre, Homologie singuliere des espaces fibres, Annals of Math. 54 (1951), pp. 425-505. As another application of 17.4, one can give a proof of the Freudenthal suspension theorem. (Compare §22.3.) CALCULUS OF VARIATIONS III. 98 Existence of Non-Conjugate Points. §.18. Theorem 17.3 gives a good description of the space viding that the points geodesic. and p O(M;p,q) pro- are not conjugate to each other along any q This section will justify this result by showing that such non- conjugate points always exist. Recall that a smooth map N - M between manifolds of the same f: dimension is critical at a point x e N f.: of tangent spaces is not 1-1. if the induced map TNx - TMf(x) We will apply this definition to the ex- ponential map TMp exp = expp: (We will assume that M M . is complete, so that exp is everywhere defined; although this assumption could easily be eliminated.) THEOREM 18.1. is critical at v(0) = v and mo(o) = X. Let a Then the map Therefore the vector field W given by yv. Obviously W(O) = 0. W(1) = .(exp v(u))I = v be a path in TMp u-. v(u) defined by is a variation through geodesics of the geodesic field along Then v e TMp. the tangent space at for some non-zero X E T(TMP)v, considered as a manifold. if and only if the v. is critical at exp Suppose that exp v to mapping PROOF: 0 from p yv exp is conjugate to p along exp v The point the geodesic t to exp*(X) TMp, such that a(u,t) = exp tv(u) given by t-* exp tv. yv -(exp tv(u))u-o is a Jacob' We also have exp dvu)(0) = exp*X = 0. u=o But this field is not identically zero since dT(o) ` TU_ 3£ (exp tv(u))I(0,0) So there is a non-trivial Jacobi field along vanishing at these points; hence p and = 7v v(u)Iu=o from p to o exp v, exp v are conjugate along 7v §18. NON-CONJUGATE POINTS is non-singular at Now suppose that exp* ent vectors X1,...,Xn in T(TMP)v. and -emu (0) with vi(o) = v Then choose paths In T)Mp dvi(u) linearly independent. a1,...,an, W1,...,Wn along Since which vanish at at both p p, u - v1(u),...,u -' vn(u) constructed as above, provide p. Since the Wi(1) = exp.(Xi) Wi are can vanish at yv vanishes This completes the proof. Let is not conjugate to theorem (§6.1). n Jacobi fields clearly no non-trivial Jacobi field along and exp v. PROOF. are n is the dimension of the space of Jacobi fields along yV, COROLLARY 18.2. p n independ- exp*(X1),..., exp.(Xn) independent, no non-trivial linear combination of the exp v. Choose v. Xi = vanishing at yv, Then 99 p e M. q Then for almost all q E M, along any geodesic. This follows immediately from 18.1 together with Sard's III. 100 §19. CALCULUS OF VARIATIONS Some Relations Between Topology and Curvature. This section will describe the behavior of geodesics in a manifold with "negative curvature" or with "positive curvature." LEMMA 19.1. for Suppose that < R(A,B)A,B > < 0 in the tangent space Then no two points of every pair of vectors A,B TMp and for every p E M. M are conjugate along any geodesic. PROOF. let J Let y V; be a geodesic with velocity vector field be a Jacobi field along and Then y. D 2 J + R(V,J)V = 0 dt2 so that DJ J> < R(V,J)V,J > > 0. , dt Therefore 2 J J> _ < 7.7 , J> + J vanishes both at DJ, J > also vanishes at throughout the interval 0 J is identically zero. REMARK. > 0 and at and to, to > 0, then the function and hence must vanish identically This implies that = dt(o) 0, This completes the proof. If A and B are orthogonal unit vectors at quantity K R(A,B)A,B > B. 0 [o,t01. J(o) A and 2 / 0. If so that CTE is monotonically increasing, and strictly Thus the function < CTE , J > so if d DJ p then the is called the sectional curvature determined by It is equal to the Gaussian curvature of the surface (u1,u2) expp(u1A + u2B) spanned by the geodesics through p with velocity vectors in the subspace spanned by A and p. 101.) B. (See for example, Laugwitz "Differential-Geometrie," TOPOLOGY AND CURVATURE §19. 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 geodesics as being the paths of light rays. Consider an observer at looking in the direction of the unit vector U towards a point A small line segment at q with length sponding to the unit vector W e TMP, p q = exp(rU). pointed in a direction corre- L, would appear to the observer as a line segment of length L(1 + - < R(U,W)U,W > + (terms involving higher powers of r)) Thus if sectional curvatures are negative then any object appears shorter than it really is. A small sphere of radius an ellipsoid with principal radii where W K1)K2,...,Kn e(1 + -K1 at q would appear to be + ...), ..., e(1 + r2 67Kn + ...) denote the eigenvalues of the linear transformation Any small object of volume - R(U,W)U. e r2 would appear to have volume v 2 v(1 + -(K1 + K2 +...+ Kn) + (higher terms)) to the "Ricci curvature" K(U,U), where K1 +...+ Kn is equal as defined later in this section.] Here are some familiar examples of complete manifolds with curvature < 0: (1) The Euclidean space with curvature (2) The paraboloid (3) The hyperboloid of rotation x2 + y2 - z2 ture (4) 0. z = x2 - y2, with curvature < 0. = 1, with curva- < 0. The helicoid x cos z + y sin z = 0, with curvature (REMARK. < 0. In all of these examples the curvature takes values arbi- trarily close to o. Of. N. V. Efimov, Impossibility of a complete surface in 3-space whose Gaussian curvature has a negative upper bound, Soviet Math., Vol. 4 (1963), pp. 843-846.) A famous example of a manifold with everywhere negative sectional curvature is the pseudo-sphere z = - 1 - x2 - y2 + with the Riemann metric induced from R3. sech 1 Jx2 + y2, z > 0 Here the Gaussian curvature has the constant value -1. No geodesic on this surface has conjugate points although two geodesics may intersect in more than one point. The pseudo-sphere gives a III. 102 non-Euclidean < it radians. CALCULUS OF VARIATIONS geometry, in which the sum of the angles of any triangle is This manifold is not complete. In fact a theorem of Hilbert states that no complete surface of constant negative curvature can be imbedded in R3. (See Blaschke, "Differential Geometric I," 3rd edn., §96; or Efimov, ibid.) However, there do exist Riemannian manifolds of constant negative curvature which are complete. (See for example Laugwitz, "Differential and Riemannian geometry," §12.6.2.) Such a manifold can even be compact; for example, a surface of genus > 2. (Compare Hilbert and Cohn-Vossen, "Geometry and the imagination," p. 259.) THEOREM 19.2 (Cartan*). Suppose that M is a simply connected, complete Riemannian manifold, and that the sectional curvature KR(A,B)A,B > is everywhere < 0. M Then any two points of desic. Furthermore, M are joined by a unique geois diffeomorphic to the Euclidean space Rn. PROOF: Since there are no conjugate points, it follows from the index theorem that every geodesic from p Theorem 17.3 asserts that the path space of a 0-dimensional to q 0(M;p,q) has index X = 0. has the homotopy type CW-complex, with one vertex for each geodesic. The hypothesis that M is simply connected implies that is connected. Thus Q(M;p,q) Since a connected 0-dimensional CW-complex must consist of a single point, it follows that there is precisely one geodesic from p q. * to See E. Cartan, "Lecons sur la Geometrie des Espaces de Riemann," Paris, 1926 and 1951. §19. Therefore, the exponential map is a global diffeomorphism. expp see that - M is one-one and is non-critical everywhere; is locally a diffeomorphism. expp so that TMp expp: expp But it follows from 18.1 that onto. 103 TOPOLOGY AND CURVATURE Combining these two facts, we This completes the proof of 19.2. is not simply connected; but is More generally, suppose that M complete and has sectional curvature flat torus S1 X S1, negative curvature.) < 0. (For example M might be a or a compact surface of genus > 2 with constant Then Theorem 19.2 applies to the, universal covering ti space M of For it is clear that M inherits a Riemannian metric M. from M which is geodesically complete, and has sectional curvature paths from p q to it follows that each homotopy class of p,q E M, Given two points < 0. contains precisely one geodesic. The fact that M is contractible puts strong restrictions on the topology of M. For example: COROLLARY 19.3. < 0 then If M is complete with <R(A,B)A,B the homotopy groups ai(M) are zero for contains no element of finite order other than the identity. i > 1; and PROOF: Clearly a1(M) M is = j(M) = contractible the cohomology group 11k(M) can be identified with the co- homology group 202 of that ai(M) of the group Hk(a1(M)) S. T. Hu "Homotopy Theory," a1(M) for 0 a1(M). (See for example pp. 200- M Hk(G) of M we have = Hk(M) Now suppose Academic Press, 1959.) contains a non-trivial finite cyclic subgroup suitable covering space Since i > 1. = 0 a1(M) = G; for G. Then for a hence k > n . But the cohomology groups of a finite cyclic group are non-trivial in arbitrarily high dimensions. This gives a contradiction; and completes the proof. Now we will consider manifolds with "positive curvature." Instead of considering the sectional curvature, one can obtain sharper results in this case by considering the Ricci tensor (sometines called the "mean curvature tensor"). CALCULUS OF VARIATIONS III. io4 fold M of a Riemannian mani- p The Ricci tensor at a point DEFINITION. is a bilinear pairing TMp K: K(U1,U2) Let defined as follows. R x TMp be the trace of the linear transforma- tion W from TMp to R(U1,W)U2 (In classical terminology the tensor K is obtained TMp. from R by contraction.) It follows easily from §9.3 that K is symmetric: K(U1,U2) = K(U2,U1). The Ricci tensor is related to sectional curvature as follows. U1,U2,...,Un be an orthonormal basis for the tangent space tures < R(Un,Ui)Un,Ui > for By definition K(Un,Un) PROOF: ( K R(Un,Ui)Un,Uj > ) . zero, we obtain a sum of TMp. is equal to the sum of the sectional curva- K(Un,Un) ASSERTION. Let i = 1,2,...,n-1. is equal to the trace of the matrix Since the n-th diagonal term of this matrix is n-1 sectional curvatures, as asserted. THEOREM 19.4 (Myers*). Suppose that the Ricci curvature K satisfies K(U,U) > (n-1)/r2 for every unit vector U at every point of M; where r Then every geodesic on M of length > Ar contains conjugate points; and hence is not is a positive constant. minimal. PROOF: Let y: [0,11 - M be a geodesic of length P1,...,Pn along parallel vector fields y point, and hence are orthonormal everywhere along Pn points along y, L. Choose which are orthonormal at one y. We may assume that so that DP V Let Wi(t) _ (sin = = at) Pi(t). L Pn , and i = 0 Then See S. B. Myers, Riemann manifolds with positive mean curvature, Math. Journal, Vol. 8 (1941), pp. 4o1-404. Duke TOPOLOGY AND CURVATURE §19. D2W. dt2 1 2E**i,Wi) - $ < Wi, _ 105 R(V,Wi)V > dt + 0 1 (sin at)2 S (a2 - L2 < R(Pn,PI)Pn,Pi > ) dt. 0 we obtain i = 1,...,n-1 Summing for 1 7- (sin n t ) 2 ((n-1 2 ) ic - L2 K(P n, P n)> dt 0 > (n-1)/r2 and L > ar E**(Wi,Wi) < 0 for some i. Now if Hence < o. y K(Pn,Pn) then this expression is This implies that the index of is positive, and hence, by the Index Theorem, that contains conju- y gate points. It follows also that dE(a(u)) u = 0. In fact if is a variation with variation vector field Wi a for is not a minimal geodesic. y Hence d2E(a(u)) 2 du = 0, CTU_ < o for small values of E(a(u)) < E(y) then u # 0. This com- pletes the proof. If M is a sphere of radius EXAMPLE. curvature is equal to (n-1)/r2. Hence 1/r2. takes the constant value It follows from 19.4 that every geodesic of length > tains conjugate points: COROLLARY 19.5. If M compact, with diameter PROOF. If Then the length of p,q e M y ar con- a best possible result. is complete, and for all unit vectors (n-1) /r2 > o < ar. K(U,U) then every sectional r K(U,U) > then M is < ar. let must be U, y < ar. be a minimal geodesic from p to q. Therefore, all points have distance Since closed bounded sets in a complete manifold are compact, it follows that M itself is compact. M of a1(M) is This corollary applies also to the universal covering space M. Since M is compact, it follows that the fundamental group finite. This assertion can be sharpened as follows. 106 CALCULUS OF VARIATIONS III. If M THEOREM 19.6. is a compact manifold, and if the Ricci tensor K of M is everywhere positive definite, 11(M;p,q) then the path space has the homotopy type of a CW-complex having only finitely many cells in each dimension. Since the space consisting of all unit vectors PROOF. is compact, it follows that the continuous function a minimum, which we can denote by of length > ar y E 0(M;p,q) (n-1)/r2 > 0. has index i = 1,2,...,k E**(Xi,Xi ) = dimensional subspace of ( i-1 i ki for 0 TOy j; from p p which vanishes E**(Xi,Xi) < 0. span a k - is negative definite. q are not conjugate along any Then according to § 16.3 there are only finitely many geodesics to q of length < ksr. desics with index REMARK. < k. Hence there are only finitely many geo- Together with §17.3, this completes the proof. 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 breaks down since, on a manifold such as the paraboloid curvature y so that X1,...,Xk and Then a In fact for each >v > k. on which E** Now suppose that the points geodesic. of length > kirr. y and such that ), , takes on Then every geodesic one can construct a vector field Xi along outside of the interval Clearly has index y K(U,U) > 0 X > 1. More generally consider a geodesic similar argument shows that on M U K(U,U) z = x2 + y2, the 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. provided by the product Sm X Sk An instructive example is of two spheres; with m,k > 2. manifold the Ricci tensor is everywhere positive definite. For this However, the sectional curvatures in certain directions (corresponding to flat tori S1 X S1 C Sm X Sk) are zero'. It is not known whether or not be remetrized so that all sectional curvatures are positive. partial result is known: can The following If such a new metric exists, then it can not be invariant under the involution from a theorem of Synge. Sm x Sk (x,y) -+ (-x,-y) of Sm x Sk. This follows (See J. L. Synge, On the connectivity of spaces §19. of positive curvature, TOPOLOGY AND CURVATURE 107 Quarterly Journal of Mathematics (Oxford), Vol. 7 (1936), pp. 316-320. For other theorems relating topology and curvature, the following sources are useful. K. Yano and S. Bochner, "Curvature and Betti Numbers," Annals Studies, No 32, Princeton, 1953. S. S. Chern, manifold, On curvature and characteristic classes of a Riemann Abh. Math. Sem., Hamburg, Vol. 20 (1955), pp. 117-126. M. Berger, Sur certaines varietes Riemanniennes a courbure positive, Comptes Rendus Acad. Sci., Paris, Vol. 247 (1958), pp. 1165-1168. S. I. Goldberg, "Curvature and Homology," Academic Press, 1962. 109 PART IV. APPLICATIONS TO LIE GROUPS AND SYMMETRIC SPACES §20. Symmetric Spaces. A symmetric space is a connected Riemannian manifold M for each p e M there is an isometry and reverses geodesics through then LEMMA 20.1 p = y(o) M which leaves M (assuming Iqlp PROOF: Let y y(t) preserves parallel vector fields along y'(t) y(t + c). = IgIp(y(t)) = Then is a geodesic and y' Iq(y(-t)) y Iq(y'(-t - c)) = = y(t + 2c). = If the vector field V is parallel along = y(0) = p q = y(c). Therefore parallel (since Ip*V(t) fixed p is a geodesic and y be a geodesic in M, and let Then IgIp(y(t)) = y(t + 2c) and y(t + 2c) are defined). More- Let and over, y'(t + c) : = y(-t). Ip(y(t)) y'(o) = q. I p i.e., if p, such that, Ip -V(-t). is an isometry) and Therefore COROLLARY 20.2. Since M y then Ip*V(o) = -V(o); Iq* Ip*(V(t)) = Ip*(V) is therefore V(t + 2c). is complete. 20.1 shows that geodesics can be indefinitely extended. COROLLARY 20.3. 1P is unique. Since any point is joined to COROLLARY 20.4. fields along field along y p by a geodesic. U,V and W are parallel vector then R(U,V)W is also a parallel If y. PROOF. If X denotes a fourth parallel vector field along y, note that the quantity < R(U,V)W,X > is constant along y. In fact, 110 APPLICATIONS IV. q = y(c), given p = y(o), p carries to consider the isometry Then q. = < R(T*UU,T*VV)T. WW,T. Xp> <R(Uq,Vq)Wq,Xq> by 20.1. is an isometry, this quantity is equal to T Since < R(UJ,VV)WW,Xp> . Thus <R(U,V)W,X> It clearly follows that vector field X. T = Iy(c/2)Ip which is constant for every parallel R(U,V)W is parallel. Manifolds with the property of 20.4 are called locally symmetric. (A classical theorem, due to Cartan states that a complete, simply connected, locally symmetric manifold is actually symmetric.) In any locally symmetric manifold the Jacobi differential equations y: R - M be a geodesic in a local- Let have simple explicit solutions. be the velocity vector at Let V = mo(o) ly symmetric manifold. p = y(o). Define a linear transformation KV: by KV(W) = R(V,W)V. Let TMp -+ TM__ TM el,...,en denote the eigenvalues of Kv. The conjugate points to p along y are the points y(ak/e) where k is any non-zero integer, and ei is any positive eigenvalue of KV. The multiplicity of y(t) as a conjugate point is THEOREM 20.5. equal to the number of tiple of PROOF: ei such that is a mul- t n/ e . First observe that KV(W),W'> KV is self-adjoint: <W,Kv(W') = This follows immediately from the symmetry relation KR(V',W')V,W R(V,W)V',W' > = Therefore we may choose an orthonormal basis Kv.(Ui) where e1, ... ,en along y KV = are the eigenvalues. by parallel translation. U1, ...,Un eiUi Mp so that , Extend the Then since for M Ui to vector fields is locally symmetric, should not be confused with the Ricci tensor of §19. SYMMETRIC SPACES 520. 111 the condition R(V,Ui)V remains true everywhere along y. eiUi = Any vector field W along y may be expressed uniquely as W(t) w1(t)U1(t) +...+ wn(t)Un(t) = . 2 Then the Jacobi equation + KV(W) = 0 takes the form Tt d2w dt2 Ui + L Since the Ui the system of eiwiUi = 0. are everywhere linearly independent this is equivalent to n equations d2wi et + eiw1 We are interested in solutions that vanish at wi(t) = Then the zeros of If ci sin (f t), wi(t) t = 0. If for some constant are at the multiples of then wi(t) = cit ei = o 0 and if ci. then wi(t) = ci sinh (41 for some constant vanishes only at This completes the proof of 20.5. t = 0. then t = n/qre-i ei < 0 ci. ei > 0 Thus if ei < 0, wi(t) 112 APPLICATIONS IV. Lie Groups as Symmetric Spaces. §21. In this section we consider a Lie group G with a Riemannian metric which is invariant both under left translations G -* G, LT: and right translation, certainly exists. as follows: RT(a) aT. = If = G Ta is commutative such a metric is compact then such a metric can be constructed G If LT(a) Let <> be any Riemannian metric on the Haar measure on Then G. new inner product <<, >> << V,W >> = G, and Let denote .t Define a is right and left invariant. .t G by on < La*RT*(V), La*RT*(W) > du(a) du(T) S GxG Then <<,>> is left and right invariant. LEMMA 21.1 If G is a Lie group with a left and right invariant metric, then G is a symmetric space. The reflection IT in any point T E G is given by the formula IT(a) = Ta-1 T. PROOF: Ie: By hypothesis and RT LT Ie*: TGe TGe Ie shows that Ie*: TGa --) TGa-1 reverses the tangent space at Finally, defining shows that each IT = reverses the tangent space of an isometry on this tangent space. Princeton, 1946.) so is certainly R1IeLa_1 is an isometry for any a E G. Since it reverses geodesics through e, Ta-1T, = the identity Ie e. I. = RTIeRT1 is an isonetry which reverses geodesics through of G is a C°° e. T. homomorphism of R into It is well known that a 1-parameter subgroup of its tangent vector at e; Now the identity = IT(a) A 1-parameter subgroup G. Define a map G by G Ie(a) Then are isometries. G is determined by (Compare Chevalley, "Theory of Lie Groups," LIE GROUPS §21. 113 LEMMA 21.2. The geodesics y in G with y(O) = e are precisely the one-parameter subgroups of G. the map Iy(t)Ie takes y(t)y(u)y(t) = for any integer n. so for some y(u) into gent vector of Hence and at e. y(nt) = y(t)n t" = n"t and y(t' + t") = y(t)n then n" +n" = is a hor..omorphism. 1-parameter subgroup. such that the tangent vector of e y n' G be a y: R geodesic through y Iv(t)Ie(a) = y(t)a y(t) t' = nit is rational so that t'/t" If Now y(u + 2t). By Lemma 20.1 = e. y(o) By induction it follows that y(u + 2t). By continuity Now let We have just seen that Let at y' y' e be the is the tan- is a 1-parameter sub- 7 This completes the proof. y' = y. A vector field X and only if be a geodesic with and some integers t y(t')y(t"). group. y: R - G Let PROOF: on a Lie group (La)*(Xb) = Xa,b are left invariant then [X,Y] for every is called left invariant G a and in b If X and Y G. The Lie algebra is also. if g of is the G vector space of all left invariant vector fields, made into an algebra by the bracket g [ 1. is actually a Lie algebra because the Jacobi identity [[x,Y],Z] + [[Y,Z],x] + [[Z,X],Y] = 0 holds for all (not necessarily left invariant) vector fields X,Y and THEOREM 21.3. Let G be a Lie group with a left and right invariant Riemannian metric. If X,Y,Z are left invariant vector fields on G then: <[X,YI,Z> R(X,Y)Z = - [[X,Y],ZI c) < R(X,Y)Z,W > PROOF: and W <X,[Y,ZI> a) b) = Z. < [X,YI,[Z,W1 > As in §8 we will use the notation X F Y for the covariant derivative of Y in the direction X. For any left invariant X the iden- tity x FX = 0 is satisfied, since the integral curves of X are left translates of parameter subgroups, and therefore are geodesics. 1- IV. 114 APPLICATIONS Therefore (X + Y) F (X + Y) = (X F X) + (X F Y) + (Y F X) + (Y F Y) is zero; hence X F Y+ Y F X = 0. X FY -Y FX = [X,Y] On the other hand Adding these two equations we obtain: by §8.5. d) 2X F Y [X,Y] = Now recall the identity Y (X,Z> = <Y F X,Z> (See §8.4.) constant. + <X,Y F Z > . The left side of this equation is zero, since <X,Z > is in this equation we obtain Substituting formula (d) 0 = < [Y,X] , Z> + Finally, using the skew commutativity of <X, [Y, Z] > [Y,X], . we obtain the required formula <[X,Y],Z> (a) By definition, R(X,Y)Z - X F (Y F Z) Substituting formula (d), - <X,[Y,Z]> _ is equal to Y F (X F Z) + [X,Y] F Z. + this becomes -[X,[Y,Z]] + -[Y,[X,Z]] + 2[[X,Y],Z] Using the Jacobi identity, this yields the required formula R(X,Y)Z (b) The formula = 7[[X,Y],Z] (c) follows from (a) and (b) . . It follows that the tri-linear function X,Y,Z -+ <IX,Y],Z > is skewsymmetric in all three variables. Thus one obtains a left invariant differential 3-form on G, representing an element of the de Rham cohomology group 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 Espaces Representatives des Groupes de Lie," Paris, Hermann, 1936.) LIE GROUPS §21. 115 The sectional curvature < R(X,Y)X,Y K [X,Y],[X,Y] > is always > 0. Equality holds if and only if [X,Y] = 0. COROLLARY 21.4. Recall that the center the set of X E such that g of a Lie algebra c [X,Y] = 0 is defined to be g for all Y E g. If G has a left and right invariant COROLLARY 21.5. metric, and if the Lie algebra g has trivial center, then G is compact, with finite fundamental group. Let This follows from Meyer's theorem (§19). PROOF: unit vector in X1 and extend to a orthonormal basis X1,...,Xn. g be any The Ricci curvature K(X1,X1) _ 2. (R(X1,Xi)X1,Xi i=1 must be strictly positive, since K(X1,X1) [X1,Xi) / for some 0 Furthermore i. is bounded away from zero, since the unit sphere in Therefore, by Corollary 19.5, the manifold G g is compact. is compact. This result can be sharpened slightly as follows. A simply connected Lie group G with left COROLLARY 21.6. and right invariant metric splits as a Cartesian product G' is compact and Rk denotes the additive Furthermore, the Lie G' x Rk where Lie group of some Euclidean space. algebra of has trivial -center. G' Conversely it is clear that any such product G' x Rk possesses a left and right invariant metric. Let PROOF. g' _ c be the center of the Lie algebra (X E g : <X, C> = be the orthogonal complement of if X,Y E g' and C E 0 c. for all Then g' [X,Y] E g'. Lie algebras. is compact by 21.5 and For is a Lie sub-algebra. then c It follows that Hence G and let CE c) [X,Y],C > = <X, 1Y,C1 > = hence g g o; splits as a direct sum splits as a Cartesian product G" G' x G"; g' ® C where of G' is simply connected and abelian, hence isomorphic 116 APPLICATIONS TV. to some Rk. (See Chevalley, "Theory of Lie Groups.") This completes the proof. THEOREM 21.7 (Bott). Let G be a compact, simply conThen the loop space 0(G) has the nected Lie group. homotopy type of a CW-complex with no odd dimensional cells, and with only finitely many X-cells for each even value of X. Thus the x-th homology groups of free abelian of finite rank for REMARK 1. an example, if G . odd, and is a. even. This CW-complex will always be infinite dimensional. is the group that the homology group REMARK 2. is zero for 01(G) As of unit quaternions, then we have seen S3 is infinite cyclic fqr all even values of i. Hi01(S3) This theorem remains true even for a non-compact group. In fact any connected Lie group contains a compact subgroup as deformation retract. (See K. Iwasawa, On some types of topological groups, Annals of Mathematics 50 (1949), Theorem 6.) PROOF of 21.7. Choose two points conjugate along any geodesic. p and Q(G;p,q) By Theorem 17.3, type of a CW-complex with one cell of dimension p to each of index q X. in G which are not q X for each geodesic from By §19.4 there are only finitely many Thus it only remains to prove that the index X. has the homotopy X %-cells for of a geodesic is always even. Consider a geodesic y V = According to §20.5 p with velocity vector starting at mo(o) e TGp = g p the conjugate points of . on y eigenvalues of the linear transformation KV : TGp - TGp , defined by KV(W) = R(V,W)V = 7[[V,w],V] Defining the adjoint homomorphism Ad V: S1 9 are determined by the 117 LIE GROUPS §21. by Ad V(W) _ [V,W] we have KV = - - (Ad V) o is skew-symmetric; that is The linear transformation Ad V Ad V(W),W' > _ (Ad V) - < W,Ad V(W') > This follows immediately from the identity 21.3a. an orthonormal basis for (4 . Therefore we can choose so that the matrix of takes the form Ad V a2 0 -a2 It follows that the composite linear transformation (Ad V)o(Ad V) has matrix 2 2 a2 Therefore the non-zero eigenvalues of KV - --(Ad V)2 are positive, and occur in pairs. It follows from 20.5 that the conjugate points of p y also In other words every conjugate point has even multiplicity. occur in pairs. Together with the Index Theorem, this implies that the index geodesic from along p to q is even. This completes the proof. ? of any 118 IV. APPLICATIONS Whole Manifolds of Minimal Geodesics. §22. So far we have used a path space o(M;p,q) based on two points p,q c M which are in "general position." However, Bott has pointed out that very useful results can be obtained by considering pairs special position. let As an example let p,q be antipodal points. desics from p to q. M be the unit sphere p,q Sn+1, in some and Then there are infinitely many minimal geo- In fact the space saa of minimal geodesics forms a smooth manifold of dimension n which can be identified with the equator Sn C Sn+1. We will see that this space of minimal geodesics provides a fairly good approximation to the entire loop space 0(Sn+1) Let M be a complete Riemannian manifold, and let points with distance p,q c M be two p(p,q) =-,r-d. THEOREM 22.1. If the space Std of minimal geodesics from p to q is a topological manifold, and if every non-minimal geodesic from p to q has index > Xo, then the relative homotopy group ai(o,Std) is zero for 0 < i < X0. It follows that the inclusion homomorphism ai(rid is an isomorphism for group ai(Sc) i < Xo - 2. is isomorphic to - ai(a) But it is well known that the homotopy ai+1(M) for all values of i. (Compare S. T. Hu, "Homotopy Theory," Academic Press, 1959, p. 111; together with §17.1.) 119 MANIFOLDS OF MINIMAL GEODESICS §22. Thus we obtain: COROLLARY 22.2. ei(od) With the same hypotheses, for 0 < i < Xo - 2. is a1+1(M) isomorphic to Let us apply this corollary to the case of two antipodal points on the (n+1)-sphere. Evidently the hypotheses are satisfied with 1`0 = 2n. Sn+1; For any non-minimal geodesic must wind one and a half times around and contain two conjugate points, each of multiplicity n, in its interior. This proves the following. (The Freudenthal suspension theorem.) COROLLARY 22.3. is isomorphic to The homotopy group ,ri(Sn) "i+1(Sn+1) for i < 2n-2. also implies that the homology groups of the loop Theorem 22.1 space 0 are isomorphic to those of fact follows from in dimensions S;d < %0 - 2. This 22.1 together with the relative Hurewicz theorem. Compare also for example Hu, p. 306. (See 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 lemma, which asserts that the condition "all critical points have index > %o" remains true when a function is jiggled slightly. Let K be a compact subset of the Euclidean space Rn; a neighborhood of let U be and let K; U f: R be a smooth function such that all critical points of f in K have index )o. LEMMA 22.4. If is "close" to _ 3xi la of I '3i < R is any smooth function which U g: f, in the sense that E a2g xT uniformly throughout _ K, for some sufficiently small constant then all critical points of (Note that the application, points.) f a2f 8x c)x i s, g in K have index > Xo. is allowed to have degenerate critical points. In g will be a nearby function without degenerate critical 120 IV. APPLICATIONS The first derivatives of PROOF of 22.4. are roughly described g by the single real valued function kg(x) on U; = >o mo1 f which vanishes precisely at the critical points of derivatives of can be roughly described by n g continuous functions U - R eg,..., eg: as follows. The second g. Let e11(x) < e2(x) <... < eg(x) 9 9 2 denote the x of g n eigenvalues of the matrix has index > ? ( ) x . if and only if the number The continuity of the functions eg Thus a critical point is negative. eg(x) follows from the fact that the X-th eigenvalue of a symmetric matrix depends continuously on the matrix*. This can be proved, for example, using the fact that the roots of a complex n vary continuously with the coefficient of the poly- polynomial of degree (Rouche's theorem.) nomial. Let denote the larger of the two numbers mg(x) Similarly let mf(x) and -ef (x). The hypothesis that all critical points of index > X0 kg(x) and -e g W. denote the larger of the corresponding numbers implies that -ef (x) > 0 whenever f kf(x) = 0. kf(x) in K have In other words for all x E K. mf(x) > 0 Let s > 0 denote the minimum of is so close to f (*) kg(x) - kf(x) for all x E K. critical point of mf on K. Now suppose that g that Then mg(x) g I < e, leg (x) - of (x)I will be positive for < e x E K; hence every in K will have index > X0. This statement can be sharpened as follows. matrices. Consider two nxn symmetric If corresponding entries of the two matrices differ by at most e, then corresponding eigenvalues differ by at most ne. This can be proved using Courant's minimax definition of the X-th eigenvalue. (See §1 of Courant, Uber die Abhangigkeit der Schwingungszahlen einer Membran..., Nachrichten, Kbniglichen Gesellschaft der Wissenschaften zu Gottingen, Phys. Klasse 1919, pp. 255-264.) Math. MANIFOLDS OF MINIMAL GEODESICS §22. 121 To complete the proof of 22.4, it is only necessary to show that the inequalities (*) will be satisfied providing that g`zi - 4i l for sufficiently small a. 2g <E and I 2 xcc1 x 6x x1c < E This follows by a uniform continuity argument which will be left to the reader (or by the footnote above ). for real valued We will next prove an analogue of Theorem 22.1 functions on a manifold. Let M -+ R be a smooth real valued function with minimum f: such that each Me = f-1[o,c] o, is compact. If the set M0 of minimal points is a manifold, and if every critical point in M - M0 has index > Xo, LEMMA 22.5. ar(M,M°) = 0 then PROOF: U C M. 0 < r < X0. for First observe that is a retract of some neighborhood Mo In fact Hanner has proved that any manifold M0 neighborhood retract. is an absolute (See Theorem 3.3 of 0. Hanner, Some theorems on absolute neighborhood retracts, Arkiv for Matematik, Vol. 1 (1950), pp. Replacing U by a smaller neighborhood if necessary, we may 389-408.) assume that each point of U is joined to the corresponding point of by a unique minimal geodesic. Let Thus U can be deformed into Mo denote the unit cube of dimension Ir r < Xp, M0 within M. and let h: (Ir,Ir) _ (M,MO) h is homotopic to a map We must show that be any map. h' with h'(Ir) C Mo. Let mum of c be the maximum of on the set M - U. f since each subset Mc - U f on h(Ir). (The function f Let 35 > 0 be the mini- has a minimum on M - U is compact.) Now choose a smooth function g: which approximates is possible by §6.8. f Mc+2s R closely, but has no degenerate critical points. To be more precise the approximation should be so close that: (1) Jf(x) - g(x)l < s for all x E Mc+2s. and This 122 APPLICATIONS IV. pact set at each critical point which lies in the com- g The index of (2) f-1[8,c+28] > x0. is g which approximates It follows from Lemma 22.4 that any f sufficiently closely, the first and second derivatives also being approximated, will satisfy (2). In fact the compact set covered by finitely many compact set f-'[8,0+281 each of which lies in a coordi- Ki, Lemma 22.4 can then be applied to each Ki. nate neighborhood. The proof of 22.5 now proceeds as follows. The function Ience the manifold points are non-degenerate, with index > ).0. g has the homotopy type of g 1(-oo,2s] (--,c+bl g is and all critical smooth on the compact region g 1[25,c+s] C f 1[s,c+2sl, -1 can be with cells of dimension attached. > X0 Now consider the map h: Since r < X0 Mc,MO Ir jr C g 1(-oo,c+s],MO h is homotopic within it follows that g1(-oo,c+s],M0 to a map Ir,Ir -y g-1 (--22812M 0 h': But this last pair is contained in MO h": within M. It follows that and U (U,MO); . can be deformed into is homotopic within h' to a map (M,MO) This completes the proof of 22.5. Ir,Ir - MO,M0. The original theorem, 22.1, now can be proved as follows. Clearly it is sufficient to prove that si(Int 0c,fd) for arbitrarily large values of a smooth manifold sad c. = o As in §16 the space Int ac(to,t1,...,tk) Int 0c as deformation retract. contains The space of minimal geodesics is contained in this smooth manifold. The energy function Int sac(t0,t1,...,tk), difficulty is that when restricted to ranges over the interval [o,oo). F. be any diffeomorphism. --' R, almost satisfies the hypothesis of E(m) the required interval E: a To correct this, let [d,c) - [o,o) 22.5. d < E < c, The only instead of §22. MANIFOLDS OF MINIMAL GEODESICS Then F ° E: Int oc(tc,t1,...,tk) satisfies the hypothesis of 22.5. R Hence si(Int cc(tp,...,tk),12d) = ni(Int fc,0d) is zero for i < Xc. This completes the proof. 123 124 APPLICATIONS IV. §23. The Bott Periodicity Theorem for the Unitary Group. First a review of well known facts concerning the unitary group. of complex numbers, with the usual Her- Let C n be the space of n-tuples mitian inner product. of all linear transformations product. is defined to be the group The unitary group U(n) S. Cn -Cn which preserve this inner is the Equivalently, using the matrix representation, U(n) group of all n x n complex matrices denotes the conjugate transpose of S such that S S* = I; where S* S. For any n x n complex matrix A the exponential of A is defined by the convergent power series expansion = I+A+ 1 A2+1A3 + ... exp A The following properties are easily verified: (1) exp (A*) (2) If A and B = exp (TAT-1) (exp A)*; T(exp A)T-1. = commute then In particular: _ (exp A)(exp B). exp (A + B) (3) (exp A)(exp -A) (4) The function exp maps a neighborhood of = I 0 in the space of n x n matrices diffeomorphically onto a neighborhood of If A is skew-Hermitian (that is if A + A* lows from (1) and (3) that exp A is unitary. = 0), Conversely if then it fol- exp A unitary, and A belongs to a sufficiently small neighborhood of it follows from (1), (3), and (4) that A + A* 0. = I. is then 0, From these facts one easily proves that: (5) (6) U(n) is a smooth submanifold the tangent space TU(n)I of the space of n x n matrices; can be identified with the space of n x n skew-Hermitian matrices. Therefore the Lie algebra g the space of skew-Hermitian matrices. of U(n) can also be identified with For any tangent vector at uniquely to a left invariant vector field on U(n). I extends Computation shows that the bracket product of left invariant vector fields corresponds to the product [A,B] = AB - BA of matrices. THE BOTT PERIODICITY THEOREM §23. Since U(n) Riemannian metric. 125 is compact, it possesses a left and right invariant Note that the function TU(n) I - U(n) exp: defined by exponentiation of matrices coincides with the function exp defined (as in §10) by following geodesics on the resulting Riemannian manifold. In fact for each skew-Hermitian matrix A the correspondence t - exp(t A) U(n) (by Assertion (2) above); defines a 1-parameter subgroup of and hence defines a geodesic. A specific Riemannian metric on U(n) Given matrices can be defined as follows. let <A,B > denote the real part of the complex A,B e g number trace (AB*) Aij$ij = . 1,3 Clearly this inner product is positive definite on This inner product on Riemannian metric on U(n). g . determines a unique left invariant g To verify that the resulting metric is also right invariant, we must check that it is invariant under the adjoint action of U(n) on g. DEFINITION of the adjoint action. Each S E U(n) determines an inner automorphism S-1 X-SX of the group U(n). U(n). Ad(S). (LSRS-1 )X The induced linear mapping TU(n)I --. TU(n)I (LSRS-1)*: is called = Thus Ad(s) is an automorphism of the Lie algebra of Using Assertion (1) above we obtain the explicit formula Ad(S)A = SAS 1 , for A E g, S E U(n). The inner product Ad(S). In fact if <A,B> Al = Ad(S)A, A1B1* = is invariant under each such automorphism B1 = Ad(S)B SAS-1(SBS 1)* then the identity SAB*S-1 = 126 1V. APPLICATIONS implies that trace (A1B1*) = trace (SAB*S 1) trace (AB*) = ; and hence that KA1,B1 > < A,B = It follows that the corresponding left invariant metric on U(n) is also right invariant. Given A c g T E U(n) so that we know by ordinary matrix theory TAT 1 that there exists is in diagonal form ia2 TAT-1 where the = Also, given any ails are real. there is S E U(n), a T E U(n) such that TST-1 = ian e where again the ails are real. Thus we see directly that exp: g -iJ(n) is onto. One may treat the special unitary group SU(n) SU(n) is defined as the subgroup of terminant 1. in the same way. consisting of matrices of de- U(n) If exp is regarded as the ordinary exponential map of matrices, it is easy to show, using the diagonal form, that det (exp A) Using this equation, one may show that the set of all matrices A etrace A = g' the Lie algebra of SU(n) , such that A + A* = and 0 trace A = 0. In order to apply Morse theory to the topology of U(n) we begin by considering the set of all geodesics in U(n) In other words, we look for all A E TU(n)1 Suppose T E U(n) A is such a matrix; be such that exp TAT-1 TAT-1 = = g is from such that and SU(n), I to -I. exp A = -I. if it is not already in diagonal form, let is in diagonal form. T(exp A)T-1 = T(-I)T-1 Then = -I THE BOTT PERIODICITY THEOREM §23. 127 so that we may as well assume that A is already in diagonal form / ia1 / eis 1 ian e if and only if A has the form exp A = -I so that / k1 in k2in. I for some odd integers k1,...,kn. Since the length of the geodesic is it ki JAI k'1 = tJ r AA*, equals and in that case, the length is + 1, to Cn as a linear map of Cn v E Cn such that Av = inv; Av = -inv. Eigen(-in), to t = is I Now, regarding observe that A is completely the space of all v E Cn splits as the orthogonal sum Eigen(in) is then completely determined by Eigen(in), which is an arbitrary subspace of Cn. from n f-n. the vector space consisting of all and Eigen(-in), Since Cn the matrix A desics in U(n) t = 0 the length of the geodesic determined by A determined by specifying Eigen(in), such that from Thus A determines a minimal geodesic if and only if each +...+ kn. such an A t -+ exp to to -I Thus the space of all minimal geo- may be identified with the space of all sub-vector -spaces of C Unfortunately, this space is rather inconvenient to use since it has components of varying dimensions. replacing U(n) by SU(n) above considerations remain valid. a1 +...+ a2m = 0 This difficulty may be removed by and setting n = 2m. with ai = + n In this case, all the But the additional condition that restricts Eigen(in) to being an arbi- trary m dimensional sub-vector-space of C. 2m This proves the following: 1 128 IV. APPLICATIONS The space of minimal geodesics from I to -I in the special unitary group SU(2m) is homeomorphic to the complex Grassmann manifold Gm(C2m), consisting of all m dimensional vector subspaces of LEMMA 23.1. C2m. We will prove the following result at the end of this section. Every non-minimal geodesic from has index > 2m+2. LEMMA 23.2. in SU(2m) to I -I Combining these two lemmas with §22 we obtain: THEOREM 23.3 (Bott). The inclusion map Gm(CM) _ induces isomorphisms of homotopy groups ci(SU(2m); 1,-I) in dimensions < 2m. Hence tti Gm(CSm) - tti+1SU(2m) for i<2m. On the other hand using standard methods of homotopy theory one obtains somewhat different isomorphisms. LEMMA 23.4. The group for tti-t U(m) U(m) = I( for i < 2m. i-1 i < 2m; ttiGm(C2m) is isomorphic to Furthermore, U(m+1) = I( i-1 U(m+2) = .. and ttj U(m) = ttj SU(m) for j X 1. PROOF. First note that for each m there exists a fibration U(m) - U(m+1)-wS2m+1 From the homotopy exact sequence -F tti g 2m+1 xi-1 U(m) -if,-, U(m+1) -1(i-1 ... S 2m+1 of this fibration we see that tti-1 U(m) - tti_1 U(m+t) for i < 2m. (Compare Steenrod, "The Topology of Fibre Bundles," Princeton, 1951, p. 35 and p. 9o.) It follows that the inclusion homomorphisms tti-1 U(m) are all isomorphisms for -+ 5i_1 U(m+t) i < 2m. -+ tti_1 U(m+2) -- ... These mutually isomorphic groups are called the 129 THE BOTT PERIODICITY THEOREM §23. (i-1)-st stable homotopy group of the unitary group. be denoted briefly by They will ai-1 U. The same exact sequence shows that, for the homomorphism i = 2m+1, n2m U is onto. n2M U(m) - a2m U(m+1) The complex Stiefel manifold is defined to be the coset space U(2m)/U(m). From the exact sequence of the fibration -'U(2m) -U(2m)/U(m) U(m) we see that ni(U(2m)/ U(m)) = for 0 i < 2m. The complex Grassmann manifold the coset space U(2m)/ U(m) x U(m). Gm(C2m) can be identified with (Compare Steenrod §7.) From the exact sequence of the fibration U(m) --F U(2m) / U(m) . Gm(C2m) we see now that niGm(C2m) for ai_l U(m) i < 2m. Finally, from the exact sequence of the fibration SU(m) -+ U(m) -+ S1 we see that = ai U(m) ni SU(m) for j / 1. This completes the proof of Lemma 23.4. Combining Lemma 23.4 with Theorem 23.3 we see that = AiGm(C2m) ni-1 U = ai-1 U(m) for 1 < i < 2m. = ni+1 SU(2m) = ni+1 U Thus we obtain: PERIODICITY THEOREM. ni+1 U for ni-1 U i > 1. To evaluate these groups it is now sufficient to observe that U(1) is a circle; so that n0 U n1 U As a check, since SU(2) = no U(1) = = n1 U(1) = Z 0 (infinite cyclic). is a 3-sphere, we have: n2 U = n3 U = n2 SU(2) n3 SU(2) = 0 =Z Thus we have proved the following result. 130 APPLICATIONS IV. THEOREM 23.5 (Bott). The stable homotopy groups rti U of the unitary groups are periodic with period 2. In fact the groups n0 U = a2 U = n4 U are zero, and the groups 3U =55U =... n1U are infinite cyclic. The rest of §23 will be concerned with the proof of Lemma 23.2. must compute the index of any non-minimal geodesic from SU(n), where n is even. matrix A e g' n x n to -I on Recall that the Lie algebra T(SU(n))I = g' consists of all I We skew-Hermitian matrices with trace zero. corresponds to a geodesic from I to A given if and only if -I the eigenvalues of A have the form isk1,...,iakn where k1,...,kn are odd integers with sum zero. We must find the conjugate points to t -h exp (tA) I along the geodesic . According to Theorem 20.5 these will be determined by the positive eigenvalues of the linear transformation g1 K : A -. g' where KA(W) R(A,W)A = = -- [[A,W),A] . (Compare §21.7.) We may assume that A is the diagonal matrix inks iskn with k1 > k2 > ... > kn. If W = (wjQ) then a short computation shows that [A,W1 = (ia(kj (A,[A,W]) = (-a2(kj - kp)2 wjQ) - kQ)wjf) , hence , THE BOTT PERIODICITY THEOREM §23. 131 and it KA(W) Now we find a basis for 1) For each (kj - kp) 2 wj f) consisting of eigenvectors of g' the matrix j < P with Ej, +1 as follows: KAY in the (j4)-th (ej)-th place and zeros elsewhere, is in g' -1 in the place, 2 and is an eigenvector corresponding to the eigenvalue 2 "(kj - kf)2 2) Similarly for each (jfl -th place and the matrix j < P in the +i E,', with +i in the (.j)-th place is an eigenvector, also with eigenvalue !(k. - kk)2 3) Each diagonal matrix in is an eigenvector with eigenvalue o. g' KA Thus the non-zero eigenvalues of with kj > kk. are the numbers (kj - k1) 2 Each such eigenvalue is to be counted twice. Now consider the geodesic y(t) Each eigenvalue exp tA. = gives rise to a series of conjugate points along e = 1 (kj - kj)2 > 0 y corresponding to the values t (See §20.5.) a/*, 2r[/,Te-, = 3a/Ve-, Substituting in the formula for t The number of such values of k . - kR j 2 equal to - 1. t e, this gives , k -k kj -2 k , k -4 k - ... 6 in the open interval Now let us apply the Index Theorem. For each , (0,1) j,P is evidently with k. > kQ 2 we obtain two copies of the eigenvalue (kj - kR)2, and hence a contri- bution of -kp 2(l to the index. Adding over all X 2 j,P _ kj > kR for the index of the geodesic As an example, if y - 1) this gives the formula (kj-kR-2) Y. is a minimal geodesic, then all of the kj 132 APPLICATIONS TV. are equal to + 1 . Hence as was to be expected. x = 0, Let Now consider a non-minimal geodesic. CASE 1. At least m+1 case at least one of the positive ki are (say) negative. ki's of the n = 2m. must be > 3, In this and we have m+1 - (-1) - 2) CASE 2. all are m of the Then one is + 1. ki > 3 are positive and m are negative but not and one is (3 - (-1) - 2) + 1 . < -3 so that m-1 m-1 > 2(m+1) = (1 - (-3) - 2) + (3 - (-3) - 2) 1 4m > 2(m+1) Thus in either case we have X > 2m+2. This proves Lemma 23.2, and therefore completes the proof of the Theorem 23.3. §24. §24. 133 THE ORTHOGONAL GROUP The Periodicity Theorem for the Orthogonal Group. This section will carry out an analogous study of the iterated loop However the treatment is rather sketchy, and space of the orthogonal group. The point of view in this section was suggested many details are left out. by the paper Clifford modules by M. Atiyah, R. Bott, and A. Shapiro, which relates the periodicity theorem with the structure of certain Clifford algebras. (See Topology, Vol. 3, Supplement 1 (1964), pp. 3-38.) Consider the vector space Rn with the usual inner product. consists of all linear maps orthogonal group 0(n) T Rn : which preserve this inner product. real n x n matrices The T Rn Alternatively 0(n) T T* such that = I. consists of all This group 0(n) considered as a smooth subgroup of the unitary group U(n); can be and therefore inherits a right and left invariant Riemannian metric. Now suppose that DEFINITION. mation J : is even. A complex structure Rn the identity n J on Rn is a linear transfor- belonging to the orthogonal group, which satisfies J2 = -I. The space consisting of all such complex structures on Rn will be denoted by St1(n). We will see presently (Lemma 24.4) that R1(n) is a smooth sub- manifold of the orthogonal group 0(n). REMARK. of 0(n) J1. Given some fixed J1 E,Q1(n) let U(n/2) be the subgroup consisting of all orthogonal transformations which commute with Then fl1(n) can be identified with the quotient space 0(n)/U(n/2). The space of minimal geodesdcs from I to -I is homeomorphic to the space n1(n) of complex on 0(n) structures on Rn. LEMMA 24.1. PROOF: The space 0(n) orthogonal matrices. the space of n x n can be identified with the group of Its tangent space g= T O(n)I skew-symmetric matrices. n x n can be identified with Any geodesic y with 134 APPLICATIONS IV. can be written uniquely as y(o) = I y(t) exp (et A) = for some A E g. Let A Since n = 2m. is skew-symmetric, there exists an element so that T E 0(n) 1 -a1 0 o a2 -a2 0 TAT-1 = o am -am o with a1,a2,...,am > 0. A short computation shows that T(exp it A)T-1 is equal to cos na1 sin aa1 0 0 -sin oat cos rta1 0 0 0 is equal to Thus exp(aA) 0 cos na2 sin aa2 0 -sin na2 cos na2 -I if and only if The inner product < A,A> Therefore the geodesic and only if If a1,a2,...,a, are odd integers. 2(a? + a2 +...+am). is easily seen to be y(t) = exp(nt A) from I to -I is minimal if a1 = a2 = ... = am = 1. y is minimal then 0 2 1 -1 0 A2 = 0 1 -1 0 T-1 T = -I hence A is a complex structure. Conversely, let J be any complex structure. nal we have J J* = I Since J is orthogo- §24. where denotes the transpose of J* this implies that J J = -I THE ORTHOGONAL GROUP J /oa T-1 TJ that a1 = ... = ah1 = 1; Thus is skew-symmetric. J Hence 1 1 -a 1 0 = a1,a2,...,am > 0 for some Together with the identity J. = -J. 1 35 and some Now the identity T. and hence that exp itJ This completes the -I. = implies J2 = -I proof. Any non-minimal geodesic from LEMMA 24.2. in O(2m) has index t -+ exp(at A) to -I Suppose that the geodesic has The proof is similar to that of 23.2. the form I > 2m-2. with A = where a1 > a2 > ... > am > 0 are odd integers. Computation shows that the non-zero eigenvalues of the linear transformation KA = - -147 (Ad A) 2 1) for each i < j the number 2) for each i < j with ai (ai + aj)2/ 4, the number # aj Each of these eigenvalues is to be counted twice. I i<j (a + aj - 2) + For a minimal geodesic we have >v = 0, as expected. I ai>aj and (ai - aj)2/ 4. This leads to the formula (ai - aj - 2) a1 = a2 = ... = am = For a non-minimal geodesic we have (3+1-2) + 0 = are 1 so that a1 > 3; so that 2m - 2. This completes the proof. Now let us apply Theorem 22.1. The two lemmas above, together with 136 APPLICATIONS TV. is a manifold imply the following. the statement that SL1(n) THEOREM 24.3 (Bott). The inclusion map 111(n) -+ o 0(n) induces isomorphisms of homotopy groups in dimensions < n-4. Hence Ai a, (n) = ni+1 0(n) for i < n-4. Now we will iterate this procedure, studying the space of geodesics from J to in f1(n); -J Assume that and so on. n is divisible by a high power of 2. Let commute *, J1,...,Jk-1 be fixed complex structures on Rn which anti- in the sense that iris for J r s. + Jsir = 0 Suppose that there exists at least one other complex structure which anti-commutes with J1,...,Jk-1' DEFINITION. Let SLk(n) denote the set of all complex structures J on Rn which anti-commute with the fixed structures J1,...,Jk-1' Thus we have SLk(n) C SLk-1 (n) C ... C SL1 (n) C 0(n) Clearly each SLk(n) is a compact set. natural to define n0(n) LEMMA 24.4. To complete the definition it is to be 0(n) Each SLk(n) is a smooth, totally geodesic** submanifold of 0(n). JR to -JR 0<R <k. The space of minimal geodesics from in SLR(n) is homeomorphic to SLR+1(n), for It follows that each component of SLk(n) For the isometric reflection of 0(n) cally carry ilk(n) is a symmetric space. in a point of SLk(n) will automati- to itself. These structures make Rn into a module over a suitable Clifford algebra. However, the Clifford algebras will be suppressed in the following presentation. ** A submanifold of a Riemannian manifold is called totally geodesic if each geodesic in the submanifold is also a geodesic in larger manifold. THE ORTHOGONAL GROUP §24. Any point in 0(n) PROOF of 24.4. J close to the identity can be where A is a "small," skew- expressed uniquely in the form exp A, close to the complex structure Hence any point in 0(n) symmetric matrix. where again A is small and J exp A; can be expressed uniquely as 137 skew. is a complex structure if and only if A J exp A ASSERTION 1. anti-commutes with J. If A PROOF: I Therefore = anti-commutes with J, then J-1A J J-1 (exp A)J exp A exp(J-1A J) exp A (J exp A)2 Conversely if -I. = = (J exp A)2 = = -A hence -I then the above computation shows that exp(J-1A J) exp A = I Since A is small, this implies that J-1A J = -A so that A anti-commutes with J. J exp A anti-commutes with the complex structures ASSERTION 2. J1,...,Jk-1 if and only if A commutes with J1,...,Jk-1' The proof is similar and straightforward. Note that Assertions I and 2 both put linear conditions on Thus a neighborhood of J in Qk(n) consists of all points A. J exp A where A ranges over all small matrices in a linear subspace of the Lie algebra This clearly implies that SLk(n) g. is a totally geodesic submanifold of 0(n). Now choose a specific point Jk E SLl(n), exists a complex structure and assume that there which anti-commutes with J1,...,Jk. J Setting J = JkA we see easily that A is also a complex structure which anticommutes with Jk. However, A comutes with J1,...,Jk-1. Hence the formula t -Jk exp(st A) defines a geodesic from Jk to -Jk in Sbk(n). Since this geodesic is minimal in 0(n), it is certainly minimal in SLk(n). IV. 138 Conversely, let ak(n). Setting y(t) y = APPLICATIONS -Jk in be any minimal geodesic from Jk to A is it follows from 24.1 that Jk exp(st A), a complex structure, and from Assertions 1,2 that A commutes with belongs to fk+1(n). This completes the proof of 24.4. The point JkA enk+1(n) REMARK. geodesic It follows easily that JA and anti-commutes with Jk. J1,...,Jk-1 which corresponds to a given has a very simple interpretation: y it is the midpoint y(z) of the geodesic. In order to pass to a stable situation, note that Ilk(n) imbedded in nk(n+n') as follows. Choose fixed anti-commuting complex structures structure determines a complex J e Slk(n) J ® Jk on Rn ®Rn' can be which anti-commutes with Ja E) J.1, for a = 1,...,k-1. Let 12k DEFINITION. spaces ak(n), denote the direct limit as with the direct limit topology. The space 0 =110 n--' w (I.e., the fine topology.) is called the infinite orthogonal group. It is not difficult to see that the inclusions 11k+1(n) give rise, in the limit, to inclusions 11k+1 THEOREM 24.5. nllk of the For each k > 0 is a homotopy equivalence. nh 0 = nh-1 111 -+ nil - n ak(n) k ' this limit map 1k+1 Thus we have isomorphisms ah-2112 = ... - "1 ah_l The proof will be given presently. Next we will give individual descriptions of the manifolds for k = 0,1,2,...,8. n0(n) is the orthogonal group. a1(n) is the set of all complex structures on Rn Given a fixed complex structure space llk(n) CnJ2 n2(n) J1 we may think of Rn as being a vector over the complex numbers. can be described as the set of "quaternionic structures" on the complex vector space C2. n/ Given a fixed J. e 112(n) we may think of 139 §24. THE ORTHOGONAL GROUP Cn/2 over the quaternions H. as being a vector space He/4 be the group of isometries of this vector space onto itself. Sp(n/4) Let Then SL 2(n) can be identified with the quotient space U(n/2)/ Sp(n/4). Before going further it will be convenient to set n = 16r. LEMMA 24.6 - (3). The space l3(16r) can be identified with the quaternionic Grassmann manifold consisting of H4r all quaternionic subspaces of Any complex structure PROOF: of H4r = R16r lows. Let into two mutually orthogonal subspaces Note that J1J2J3J1J2J3 determines a splitting J3 E a3(16r) and V2 as fol- is an orthogonal transformation with square J1J2J3 equal to V1 Hence the eigenvalues of + I. V1 C R16r be the subspace on which J1J2J3 J1J2J3 equals + I; are + 1. and let V. be the orthogonal subspace on which it equals -I. Then clearly R16r = V1 ® V2. Since J1J2J3 commutes with J1 and J2 it is clear that both V1 and V2 are closed under the action of Conversely, given the splitting H4r = V1 (D V2 we can define orthogonal quaternionic subspaces, J1 and J2. into mutually J3 E a3(16r) by the identities J3IV1 J3IV2 = = -J1J2lV1 J1J21V2 This proves Lemma 24.6 -(3). The space a3(16r) varying dimension. of largest dimension: spaces of H4r. is awkward in that it contains components of It is convenient to restrict attention to the component namely the space of 2r-dimensional quaternionic sub- Henceforth, we will assume that this way, so that J3 has been chosen in dimes V1 = dimes V2 = 2r. can be identified The space n4(16r) to with the set of all quaternionic isometries from V1 V2. Thus Sl4(16r) is diffeomorphic to the symplectic group Sp(2r). LEMMA 24.6 - (4). PROOF: Given J4 E n4(16r) commutes with J1J2J3 . Hence J3J4 note that the product maps V1 J3J4 to V2 (and V2 antito V1). 140 APPLICATIONS IV. Since commutes with J1 J3J4 and J3J4fV1 T V1 : V1 : - V2 Conversely, given any such isomorphism is a quaternionic isomorphism. -. V2 we see that J2 it is easily seen that is uniquely determined by the J4 identities: J4IV1 = J31 T J4IV2 = -T-1 J3 This proves 24.6 - (4). The space Sl5(16r) can be identified with the set of all vector spaces W C V1 such that (1) W is closed under J1 (i.e., W is a complex vector space) and (2) V1 splits as the orthogonal sum W ® J2 W. LEMMA 24.6 - (5). Given J5 ea5(16r) PROOF: commutes with J1J2J3 note that the transformation J1J4J5 and has square itself; and determines a splitting of Let W C V1 spaces. Since J2 Thus + I. V1 into V1 into two mutually orthogonal sub- be the subspace on which J1J4J5 anti-commutes with J1J4J5, maps J1J4J5 coincides with it follows that precisely the orthogonal subspace, on which J1J4J5 J2W C V1 equals -I. + I. is Clearly J1W = W. Conversely, given the subspace W, it is not difficult to show that J5 is uniquely determined. REMARK. morphisms of V1 If U(2r) C Sp(2r) denotes the group of quaternionic auto- keeping W fixed, then the quotient space Sp(2r)/ U(2r) can be identified with a5(16r). LEMMA 24.6 - (6). The space Sl6(16r) can be identified with the set of all real subspaces X C W such that W splits as the orthogonal sum X ® J1X. PROOF. Given J6 e a6(16r) commutes both with J1J2J3 itself. Since note that the transformation J2J4J6 and with J1J4J5. (J2J4J6)2 = I, it follows that Hence J2J4J6 J2J4J6 maps W into determines a THE ORTHOGONAL GROUP §24. 141 Let splitting of W into two mutually orthogonal subspaces. equals subspace on which J2J4J6 subspace on which it equals +I. X C W be the Then J1X will be the orthogonal -I. Conversely, given X C W, it is not hard to see that J6 is unique- ly determined. denotes the group of complex automor- If O(2r) C U(2r) REMARK. phisms of W keeping X fixed, then the quotient space U(2r)/ O(2r) can be identified with a 6(16r). can be identified LEMMA 24.6 - (7). The space Sl7(16r) with the real Grassmann manifold consisting of all real subspaces of X = R2r. PROOF: Given J7, anti-commuting with J1, ... ,J6 J1J6J7 commutes with J1J2J3, square +I. Thus J1J6J7 orthogonal subspaces: J1J6J7 equals X1 with J1J4J5, and with J2J4J6; determines a splitting of X (where J1J6J7 equals Conversely, given X1 C X -I). note that +I) and has into two mutually and X2 (where it can be shown that J7 is uniquely determined. This space 117(16r), like J1 3(16r), has components of varying dimension. Again we will restrict attention to the component of largest dimen- sion, by assuming that dim X1 = dim X2 = r. Thus we obtain: ASSERTION. The largest component of a7(16r) is diffeomorphic to the Grassmann manifold consisting of r-dimensional subspaces of R2r. LEMMA 24.6 - (8). The space fl8(16r) can be identified with the set of all real isometries from X1 to X2. PROOF. If J8 e j8(16r) commutes with J1J2J3, J1J4J5, Hence J7J8 determines maps J8 X1 then the orthogonal transformation and J2J4J6; isomorphically onto J7`T8 but anti-commutes with J,J6J7. X2. Clearly this isomorphism uniquely. Thus we see that d18(16r) is diffeomorphic to the orthogonal IV. 142 APPLICATIONS group* 0(r). Let us consider this diffeomorphism Sl8(16r) - 0(r), the limit as r It follows that Sl8 -+ -. and pass to is homeomorphic to the infinite Combining this fact with Theorem 24.5, we obtain the orthogonal group O. following. THEOREM 24.7 (Bott). The infinite orthogonal group 0 has the same homotopy type as its own 8-th loop space. ai 0 is isomorphic to the homotopy group If Sp =114 Hence "i+8 0 for i > 0. denotes the infinite symplectic group, then the above argument also shows that 0 has the homotopy type of the 4-fold loop space and that 0000 Sp, 0000 O. Sp has the homotopy type of the 4-fold loop space The actual homotopy groups can be tabulated as follows. i modulo 8 ni Sp ni O Z2 Z2 0 2 0 0 3 Z Z 4 0 5 0 Z2 Z2 0 1 0 6 0 0 7 Z Z The verification that these groups are correct will be left to the reader. (Note that Sp(1) is a 3-sphere, and that S0(3) is a projective 3-space.) The remainder of this section will be concerned with the proof of Theorem 24.5. It is first necessary to prove an algebraic lemma. Consider a Euclidean vector space V with anti-commuting complex structures * J1,...,Jk. For k > 8 it can be shown that dlk(16r) In fact any additional complex structures is diffeomorphic to Slk_8(r). J9,J10,...,Jk on R16r give rise to anti-commuting complex structures J8J9, J8J10, J8J111...,J81k X1; and hence to an element of fl k_8(r). will be sufficient to stop with k = B. on However, for our purposes it §24. THE ORTHOGONAL GROUP V is a minimal DEFINITION. 143 (J1,...,Jk)-space if no proper, non- trivial subspace is closed under the action of and J1,..., Jk. Two such minimal vector spaces are isomorphic if there is an isometry between them which commutes with the action of LEMMA 24.8 J1,...,Jk. For k 3 (mod 4), any vector spaces are isomorphic. (Bott and Shapiro). two minimal (J1,...,Jk) 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 re als, the complex numbers or the quaternions. For k = 3 a minimal space is still a 1-dimensional vector space over the quaternions. J3 is equal to Clearly any two such are isomorphic. However, there are two possibilities, according as +J1J2 This gives two non-isomorphic minimal -J1J2. or spaces, both with dimension equal to For k = 4 J3J4 mapping H to k = 5,6 For complex structures spaces. k = 7 For 4. H and Call these a minimal space must be isomorphic to The dimension is equal to H'. H'. H ® H', we obtain the same minimal vector space J5,J6 with 8. H ® H'. merely determine preferred complex or real sub- we again obtain the same space, but there are two possibilities, according as J7 is equal to +J1J2J3J4J5J6 or to -J1J2J3J4J5J6. Thus in this case there are two non-isomorphic minimal vector spaces; call these For with J7J8 k = 8 k > 8 periodically. L and L'. a minimal vector space must be isomorphic to L ® L', mapping L onto For The L'. The dimension is equal to 16. it can be shown that the situation repeats more or less However, the cases k < 8 will suffice for our purposes. Let mk denote the dimension of a minimal (J1,...,Jk)-vector space. From the above discussion we see that: m0 = 1, m1 = 2, m2 = m3 = 4, m4 = m5 = m6 = m7 = 8, m8 = 16. For k > 8 it can be shown that mk = 16mk-8. REMARK. These numbers mk are closely connected with the problem of constructing linearly independent vector fields on spheres. example that J1,...,Jk Suppose for are anti-commuting complex structures on a vector APPLICATIONS IV. 144 space V of dimension for each unit vector u E V the dicular to each other and vector fields on an Here rmk. to can be any positive integer. r k vectors u. Thus we obtain k fields on a (16r-1)-sphere; and so on. are perpenindependent 3 vector (8r-1)-sphere; 7 vector fields on an (4r-1)-sphere; (Compare linearly For example we obtain (rmk 1)-sphere. fields on a and Radon. J1u, J2u,..., Jku Then 8 vector These results are due to Hurwitz B. Eckmann, Gruppentheoretischer Beweis des Satzes von Hurwitz-Radon..., Commentarii Math. Hely. Vol. 15 (1943), pp. 358-366.) J. F. Adams has recently proved that these estimates are best possible. k V 2 (mod 4). PROOF of Theorem 24.5 for minimal geodesics from J at of 11k(n) Let T to -J in 11k(n). 1) A is skew 2) A anti-commutes with J 3) A commutes with J1,".,Jk-1' denote the vector space of all such matrices corresponds to a geodesic t - J exp (at.4) from J its eigenvalues are all odd multiples of Each such A E T KA: Recall that the tangent space J A where consists of all matrices J We must study non- T - T. can compute A. to A given A E T -J if and only if i. determines a self-adjoint transformation is a totally geodesic submanifold of 0(n), Since a k(n) we KA by the formula KAB just as before. = -1 [A,[A,B1] _ (-A2B + 2ABA - BA2)/4 , We must construct some non-zero eigenvalues of KA so as to obtain a lower bound for the index of the corresponding geodesic t - J exp(irt A) . Split the vector space Rn as a direct sum M1 ® M2 ® ... ® Ms of mutually orthogonal subspaces which are closed and minimal under the action of J1,...,Jk-1, J and A. all equal, except for sign.* Then the eigenvalues of A on Mh must be For otherwise Mh would split as a sum of * We are dealing with the complex eigenvalues of a real, skew-symmetric transformation. Hence these eigenvalues are pure imaginary; and occur in conjugate pairs. THE ORTHOGONAL GROUP §24. A; eigenvalues of AIMh; where Now note that is a complex structure on Mh which J' = ah1JAIMh; be zero for (J1,...,Jk-1,J,J')k + Since 3 (mod 4) 1 are mutually isomorphic. M1,M2,...,Ms with h h,j we can construct an eigenvector j B: Rn _Rn of the linear transformation BIM, Mh is Thus Hence the dimension of Mh is mk+1 For each pair + iah be the two are odd, positive integers. a1,...,as anti-commutes with J1,...,Jk-1, and J. we see that Let and hence would not be minimal. eigenspaces of minima].. 145 T - T as follows. Let KA: Let BIMh be an isometry from Mh to Mj f / h,j. which satisfies the conditions BJa = J0B BJ = -JB and BJ' dicates that we have changed the sign of exists by Finally let 24.8. <Bv,w> B We claim that eigenvalue B J. BIMh. Since T. w E Mj BIMh com- It follows easily that also commutes with J1,...,Jk-1 BIMj where the bar in- Such an isomorphism It is also clear that and anti-commutes with Thus Mj; for v E Mh, is skew-symmetric. the negative adjoint J. J to Mj. be the negative adjoint of Kv,- Bw> = mutes with J1,...,Jk-1 commutes with BIMj on . B belongs to the vector space Proof that it is clear that +J'B = is an isomorphism from Mh BIMh In other words a = 1,...,k-1; for and anti- B E T. KA is an eigenvector of For example if (ah + aj)2/4. (KAB)v v e Mh (-A2B + 2ABA = - corresponding to the then BA2)v 2ajBahv + Bahv) _ 7 -IT (aj + ah) 2 By and a similar computation applies for Now let us count. by v E Mj. The number of minimal spaces Mh C Rn is given For at least one of these the integer s = n/mk+1' For otherwise we would have a minimal geodesic. (always for = > 3. This proves the following 2 (mod 4)): k ASSERTION. > (3+1)2/4 ah must be 4. KA has at least The integer s-1 eigenvalues which are s = n/mk+1 tends to infinity with n. 146 APPLICATIONS IV. Now consider the geodesic of t e2 Each eigenvalue -+ J exp(at A). gives rise to conjugate points along this geodesic for KA Thus if by 20.5. t = e-1, 2e-1, 3e-1,... then one obtains at e2 > 4 Applying the index theorem, this proves least one interior conjugate point. the following. The index of a non-minimal geodesic from J ASSERTION. to -J in is > n/mk+1- 1. Slk(n) It follows that the inclusion map ak+1(n) -+ o nk(n) induces isomorphisms of homotopy groups in dimensions number tends to infinity with as n -+ co, it follows that the inclusion map i : and But it can be shown a Dk have the homotopy type of a CW-complex. fore, by Whitehead's theorem, it follows that nilk(n) Thus ing subspace fk+1(n) has an infinite cyclic fundamental has infinitely many components, while the approximathas only finitely many. To describe the fundamental group f as follows. Let k 0 2 (mod 4). The difficulty in this case may k e 2 (mod 4). be ascribed to the fact. that dlk(n) group. There- is a homotopy equivalence. i This completes the proof of 24.5 providing that PROOF of 24.5 for This -i a S).k induces S?,k+1 isomorphisms of homotopy groups in all dimensions. that both SZk+1 < n/mk+1 - 3. Therefore, passing to the direct limit n. J1,...,Jk-1 dlk(n) : we construct a map allk(n) -o S1 C C be the fixed anti-commuting complex struc- ture on R. Make Rn into an (n/2)-dimensional complex vector space by defining iv = J1 J2 ... Jk-1v for that v e Rn; where i2 = -1, i = 47-1 and that E C. The condition J1,J21...IJk-1 k 2 (mod 4) commute with i. Choose a base point J e llk(n) . For any J' e Slk(n) composition J-1J' commutes with i. Thus J-1J' guarantees note that the is a unitary complex linear transformation, and has a well defined complex determinant which will be denoted by f(J'). THE ORTHOGONAL GROUP §24. 147 Now consider a geodesic J exp(sctA) t from J to -J Since A in 1Zk(n). commutes with i = J1J2 ... Jk-1 also as a (compare Assertion 2 in the proof of 24.4) we may think of A In fact A complex linear transformation. trace of A is a pure imaginary number. is skew-Hermitian; Now f(J exp(stA)) = determinant (exp(stA)) = Thus f ent trace A maps the given geodesic into a closed loop on pletely determined by the trace of S1 which is com- It follows that this trace is in- A. n(11k(n);J,-J). variant under homotopy of the geodesic within the path space The index is closed under the action of h, J1,...,Jk-1,J, and iah. A; and is minimal. J1,...,Jk-1, Therefore the trace of A Now for each h X j can have only For otherwise Mh would split into eigenspaces. Thus AIMh coincides with ahJlJ2 ... Jk-1IMh. the action of where each Mh M1 ® ... ® Mr the complex linear transformation AIMh one eigenvalue, say As of this geodesic can be estimated as follows. % before split Rn into an orthogonal sum Thus for each hence the and Since Mh is minimal under its complex dimension is J; is equal to mk/2. i(a1+...+ar)mk/2. an eigenvector B of the linear transforma- tion B -+ KAB = (-A2B + 2ABA - BA2) /4 can be constructed much as before. Since Mh and Mj are (J1,...,Jk-1,J)- minimal it follows from 24.8 that there exists an isometry BIMh : and anti-commutes with J. which commutes with J1,...,Jk-1 the negative adjoint of BIMh; Mh-> Mj and let BIM1 be zero for Let BIMj e / h,j. Then an easy computation shows that KAB Thus for each ah > aj Since = (ah - aj) 2B/4 we obtain an eigenvalue (ah - aj)2/4 each such eigenvalue makes a contribution of towards the index x, we obtain the inequality (ah- aj - 2) 2),> ah > aj for (ah - aj)/2 - KA. 1 be 148 APPLICATIONS TV. Now let us restrict attention to some fixed component of That is let us look only at matrices A c Q ak(n). trace A = icmk/2 such that where is some constant integer. Thus the integers al,...ar 1) a, a a2 = ... = ar 2) al +...+ ar = c, 3) Max lahi > 3 satisfy (mod 2), (since exp(aA) = -I), and (for a non-minimal geodesic). h Suppose for example that some the positive ah and -q ah is equal to the sum of the negative p-q = hence 2p > r + c. ah. Thus , Now ah>a3 hence p+q>r c, (ah - a - 2) > 2X > Let p be the sum of -3. 4X > 2p > r + c; where follows that the component of ah>o = p tends to infinity with r = n/mk C 11k(n) (ah - (-3) - 3) n. It is approximated up to higher and higher dimensions by the corresponding component of Sk+l(n), as n - C . Passing to the direct limit, we obtain a homotopy equivalence on each component. This completes the proof of 24.5. 149 APPENDIX. THE HOMOTOPY TYPE OF A MONOTONE UNION The object of this appendix will be to give an alternative version for the final step in the proof of Theorem 17.3 (the fundamental theorem of Morse theory). Given the subsets C saa2 C ... and given the information that each o = o(M;p,q), space a1 oao C of the path a o i has the homotopy type of a certain CW-complex, we wish to prove that the union 0 also has the homotopy type of a certain CW-complex. More generally consider a topological space X0 C X1 C X2 C ... X X and a sequence To what extent is the homotopy type of of subspaces. Xi? determined by the homotopy types of the It is convenient to consider the infinite union X£ = X0 x[0,1] v X 1 X (1,2] u X2 X(2,31 u ... This is to be topologized as a subset of X x R. We will say that DEFINITION. the sequence (Xi) EXAMPLE 1. X£ - X, : Suppose that each point of X X and that Xi, p defined by is a homotopy equivalence. p(x,T) = x, some is the homotopy direct limit of X if the projection map is paracompact. lies in the interior of Then using a partition of unity one can construct a map f : X- R f(x) > i+1 so that pondence for x - (x,f(x)) x Xi, f(x) > 0 for all x. Now the corres- maps X homeomorphically onto a subset of is clearly a deformation retract. and X and Therefore p X. which is a homotopy equivalence; is a homotopy direct limit. EXAMPLE 2. with union X. Let X be a CW-complex, and let the Xi be subcomplexes Since p : X£ - X induces isomorphisms of homotopy groups in all dimensions, it follows from Whitehead's theorem that X direct limit. is a homotopy APPENDIX 150 The unit interval EXAMPLE 3. [0,1] limit of the sequence of closed subsets [o] is not the homotopy direct v [1/i,1]. The main result of this appendix is the following. Suppose that X is the homotopy direct and Y is the homotopy direct limit Y be a map which carries each Let f: X {Yi). into Yi by a homotopy equivalence. Then f THEOREM A. limit of of [X.1 Xi itself is a homotopy equivalence. Assuming Theorem A, the alternative proof of Theorem 17.3 can be given as follows. Recall that we bad constructed a commutative diagram saa, S2ao C Ko of homotopy equivalences. C K1 Since 1a C saa2 C ... C K2 C ... a = U o i and K = U Ki are homotopy direct limits (compare Examples I and 2 above), it follows that the limit mapping 0- K is also a homotopy equivalence. PROOF of Theorem A. Define f£ : XX -* YY It is clearly sufficient to prove that CASE 1. is a homotopy equivalence. f£ is homotopic to the identity. fi : Xi-.. Yi We must prove f£ is a homotopy equivalence. REMARK. that f£(x,t) = (f(x),t). Suppose that Xi = Yi and that each map (obtained by restricting f) that by Under these conditions it would be natural to conjecture f. must actually be homotopic to the identity. However counter- examples can be given. For each n let Xn Xn be a one-parameter family of mappings, with hn Define the homotopy h : Xz - X& fn, hn = identity. HOMOTOPY OF A MONOTONE UNION 151 as follows (where it is always to be understood that and 0 < t < 1, n = 0,1,2,...). (hu(x),n+2t) hu(x,n+t) (h(3-4t)u (x),n+1) 0,,n+ for 0 < t < z for z < t < -T for 3 < t < 1 (x),n+1) Taking u = 0 to fE. this defines a map The mapping XE : XE h1 h0 : XE which is clearly homotopic X£ on the other hand has the following properties: h1(x,n+t) = (x,n+2t) for 0 < t < z h1(x,n+t) e 71, +1 for 2 < t < 1 We will show that any such map homotopy inverse h1 x[n+11 can be defined by the formula g : XE - XE (x,n+2t) ( In fact a is a homotopy equivalence. 0 < t < - g(x,n+t) = j l h (x,n 2- t) 2 1 - <t<1 This is well defined since h1(x,n+2) = h1(x,n+1) = (x,n+1) Proof that the composition h1g of XE. . is homotopic to the identity map Note that h1g(x,n+t) = Define a homotopy Hu : XE (x,n+4t) 0 < t < 4 h1(x,n+2t) 4 < t < 2 h1 (X,ri 2 - t) 2<t<1 XE as follows. h1g(x,n+t) Hu(x,n+t) = and h1(x,n+1-u) For 0 < u < z let for 0 < t < (1-u)/2 for z+u < t < 1 for (1-u)/2 < t < z + U. APPENDIX 152 This is well defined since h1g(x,n+(1-u)/2) Now h1g and is equal to Ho h1g(x,n+2+u) = h1(x,n+1-u). is given by Hiz 0 < t < 4 1(x,n+4t) Hi(x,n+t) z = = < t < 1 (x,n+1) Clearly this is homotopic to the identity. is homotopic to the identity; and a completely analogous h1g Thus argument shows that is homotopic to the identity. gh1 This completes the proof in Case 1. Now let CASE 2. and Y be arbitrary. X gn : Yn -+ Xn be a homotopy inverse to fn. gri Yn For each n let Note that the diagram Xn tin Iin gYl+ 1 Xn+1 Yn+1 (where in and in denote inclusion maps) is homotopy commutative. In fact 1ngn ti gnifn+tingn Choose a specific homotopy and define G : Y£ X£ hu gn+tjnfngn - gn+1jn = Yn--' Xn+1 : with ho = ingn, h, gn+1Jn; by the formula (gn(y),n+2t) 0 < t < G(y,n+t) (h2t-1(y),n+1) We will show that the composition Gf£ Let X£ denote the subset of T < n. (Thus sition Gf£ identity. X£ = X0 x[o,11 carries X£ -+ X. is a homotopy equivalence. consisting of all pairs ... with (x,T) Xn_1 x [n-1 n] u Xn x[n].) The compo- X2 into itself by a mapping which is homotopic to the In fact X£ mapping GfE XE : < t < 1 contains Xn x[n] restricted to Xn x[n] hence is homotopic to the identity. as deformation retract; and the can be identified with gnfn, and Thus we can apply Case 1 to the sequence HOMOTOPY OF A MONOTONE UNION (X£n} and conclude that This proves that argument shows that f£ Gf£ f. is a homotopy equivalence. has a left homotopy inverse. f£G : Y£ - Y. has a right homotopy inverse. equivalence (compare page 22) 153 This proves that f£ is a homotopy and completes the proof of Theorem A. Suppose that X is the homotopy direct has the homotopy type of a OW-complex, then X itself has the homotopy COROLLARY. limit of (Xi). If A similar is a homotopy equivalence, so that each Xi type of a OW-complex. The proof is not difficult. 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