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The structure of manifolds of nonnegative sectional curvature Cameron, Christy 2007

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The Structure of Manifolds of Nonnegative Sectional Curvature by Christy Cameron B.Sc.(Hon.), Dalhousie University, 2005 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F Master of Science in The Faculty of Graduate Studies (Mathematics) The University Of British Columbia August, 2007 © Christy Cameron 2007 Abstract Understanding the structure of a Riemannian Manifold based on information about its sectional curvature is a challenging problem which has received much attention. According to the Soul Theorem any complete noncom-pact Riemannian manifold M of nonnegative sectional curvature contains a compact totally geodesic submanifold called the soul of M. Furthermore, the manifold is diffeomorphic to the normal bundle of the soul. This is a beautiful structural result which provides a significant contribution to the classification of Riemannian manifolds. In this paper we present a complete proof of the Soul Theorem which draws upon the theory and techniques developed over the years since its original proof in 1972. The proof relies heavily upon results from Comparison Geometry and the theory of convex sets. ii Table of Contents Abstract i i Table of Contents i i i List of Figures iv Acknowledgements v 1 Introduction 1 1.1 The Soul Theorem - Significance 3 1.1.1 Existence of a Soul 3 1.1.2 Generalization of Simple Points 4 2 Comparison Theorems 6 2.1 Rauch Comparison Theorem 6 2.2 Toponogov's Theorem 18 3 Preliminaries for the Proof of The Soul Theorem 21 4 Proof of The Soul Theorem 34 4.1 Constructing the Soul 34 4.2 Finding a Diffeomorphism 45 5 Final Comments 49 5.1 Properties and Examples of a Soul 49 5.2 The Soul Conjecture 50 5.3 Recent Research Developments 51 Bibliography 53 i i i List of Figures 2.1 Smooth variation of 7 7 2.2 A geodesic triangle 19 2.3 A geodesic hinge 19 3.1 Geodesic Cone 23 3.2 Construction of H1 26 3.3 Proof that d(q, Bli2) < t2 - h 30 3.4 Hypersurface and corresponding geodesic cone 32 4.1 Parallel Translation 36 4.2 Case 1 38 4.3 Case II 40 4.4 F(s,t) : [a,b] x [0,/] C 42 4.5 <p{s,t) : [a,b] x [0,1] -» C 43 5.1 Examples of the non-uniqueness of souls 49 iv Acknowledgements I wish to express my sincere thanks to my supervisor Dr. Ailana Fraser. I am grateful to her for introducing me to this subject and for all her help throughout this project. Her encouragement, guidance, and teaching, during my time at UBC are greatly appreciated. I also thank Dr. Jingyi Chen for the many helpful comments he made regarding an earlier draft of this paper. Finally I wish to thank my family and in particular my parents, for always being there with endless support and encouragement in all that I do. Chapter 1 Introduction Determining the structure of Riemannian manifolds based on information about sectional curvature has always been a central problem in Rieman-nian Geometry. Early investigations focused on the simplest case, namely those manifolds which possessed constant sectional curvature. A tremen-dous breakthrough came in the late 1920's with the establishment of the fact that for every K 6 K there exists exactly one (up to isometry) simply connected complete Riemannian manifold of constant sectional curvature K [1]. For manifolds of dimension n these are: Euclidean space (R n ) when K = 0, the n-Sphere (Sn) of radius when K > 0 and n- dimensional hyperbolic space (Hn) of curvature K < 0. To simplify later computations, model spaces based on this classification were defined with K = 0, K = — 1, K = 1. Through suitable adjustments to the metric all spaces with constant K are equivalent to the model space equipped with K of the same sign. Following the classification of manifolds admitting a metric of constant sectional curvature, the next step was to attempt to classify arbitrary Rie-mannian maniflods. Since much was known about the structure of manifolds of constant sectional curvature, a natural approach was to seek a classifica-tion of these manifolds through comparison with model spaces. The hope was that if information such as bounds on K were known it would be pos-sible to show that the manifold has a structure "close to" that of one of the model spaces. This technique forms the basis for the branch of Rieman-nian Geometry known as Comparison Geometry. Researchers in this area of mathematics have made immense contributions to the understanding of the structure of Riemannian manifolds and this remains an active area of research today. Since the classification of arbitrary Riemannian manifolds is a topic ex-tremely rich in theory and results, it is not possible to provide a complete overview of the history in this paper. For our purposes we will restrict our attention to complete noncompact Riemannian manifolds of nonnegative curvature. Myers and Bonnet began the classification of this family of manifolds when they developed their much celebrated Bonnet-Myers Theorem which 1 Chapter 1. Introduction states that any Riemannian manifold with sectional curvature K satisfying K > S > 0 has diameter less than or equal to This theorem implies that when the sectional curvatures are bounded away from zero then the manifold is necessarily compact. Thus the only noncompact complete man-ifolds of nonnegative curvature must contain points at which the sectional curvature gets arbitrarily close to zero. By exploiting the Gauss-Bonnet formula specified for surfaces in R2, Cohn-Vossen determined the structure of all such manifolds in dimension 2. In particular, any 2-dimensional non-compact complete manifold with K > 0 is either diffeomorphic to M.2 or is flat [4]. This was a significant finding but a new approach would be required to classify manifolds of higher dimension. In 1969, Gromoll and Meyer [6] discovered a new technique which al-lowed them to determine the structure of Riemannian manifolds when the curvature is strictly positive. Their remarkable result is that any such n-dimensional noncompact manifold is diffeomorphic to 1™. A crucial step in their novel approach which allowed for classification involved associat-ing to each point in the manifold a totally convex set. A set is said to be totally convex if whenever a geodesic has endpoints in the set then the en-tire geodesic is contained in the set. The techniques used by Gromoll and Meyer could not be generalized from the case of positive curvature to that of nonnegative curvature. But shortly thereafter, Cheeger and Gromoll [4] discovered that through the use of Toponogov's Theorem it is possible to construct totally convex sets associated to every point in a complete man-ifold carrying the relaxed condition that K > 0. Toponogov's Theorem is a result from Comparison Geometry which allowed Cheeger and Gromoll to draw geometric conclusions regarding a manifold having K > 0 through comparison with ordinary Euclidean space. Toponogov's Theorem will be discussed in detail in Chapter 2. This construction lead immediately to the discovery of what is now known as The Soul Theorem. This is a beautiful structural result and so we state it below as a separate theorem. Theorem 1.0.1. (The Soul Theorem) Every complete noncompact Riemannian manifold of nonnegative sectional curvature contains a compact totally geodesic subnianifold known as the soul of a manifold. Furthermore, the manifold is diffeomorphic to the normal bundle of any of its souls. Cheeger and Gromoll are responsible for the discovery of the Soul Theo-rem but their proof exhibited only a homeomorphism between the manifold and its soul. Their proof was extremely technical and they included a brief 2 Chapter 1. Introduction description on how to "smooth out" the homeomorphism to obtain a dif-feomorphism between the structures. Two years later, Walter A. Poor used his own new approach to explicitly construct the necessary diffeomorphism establishing the Soul Theorem in its entirety [15]. The remainder of this paper will focus on the Soul Theorem and in par-ticular we will consolidate information from various sources to provide a complete unified proof of the theorem. Although technical at times, the proof contains many beautiful geometric arguments and highlights in par-ticular the usefulness of the study of totally convex sets in understanding the structure of manifolds. Following the proof of the theorem we will re-turn to a brief discussion of the implications of the Soul Theorem on the classification of noncompact manifolds of nonnegative curvature. 1.1 The Soul Theorem - Significance The Soul Theorem is remarkable not only for its significance in the classi-fication of noncompact manifolds of nonnegative curvature. The statement of the theorem itself is quite surprising and in addition it provides a gener-alization of structures present in manifolds of strictly positive curvature. 1.1.1 Existence of a Soul According to the Soul Theorem, every complete noncompact manifold of nonnegative curvature contains a totally geodesic submanifold. As demon-strated in the discussion which follows, this is quite a remarkable statement since in general an arbitrary Riemannian manifold will not contain such a manifold. By definition, a totally geodesic submanifold 5 is a submanifold in which every geodesic in S is also a geodesic in the surrounding manifold M . Consider the Levi-Civita connection V M on M and the induced connection V 5 on S and suppose X and Y are vector fields tangent to S and extended arbitrarily to M. We may write v^y = (v^r)T + (v^y)x the sum of the tangential and normal components respectively. We define the second fundamental form II to be the map II : T(TS) x T(TS) -» T(NS) satisfying 3 Chapter 1. Introduction II(X,Y) = (V^Y)± where TS is the tangent bundle to S and TV.? is the normal bundle to S. The second fundamental form is a symmetric bilinear form. We may rewrite the previous decomposition in terms of the second fundamental form and express the connection as v#y = (v^r)T + II(X,Y) Since X and Y are arbitrary extensions of vector fields tangent to S we have V%Y = (X7SXY) + II(X,Y) This is known as the Gauss formula and it illustrates the fact that the second fundamental form is a measure of the difference between the connections [11]. From the Gauss formula it is also clear that it is not necessarily true that geodesies in S are geodesies in the surrounding manifold M. This is exactly the property that characterizes a totally geodesic submanifold and it holds precisely when the second fundamental form vanishes. Therefore, totally geodesic submanifolds are also characterized by the vanishing of the second fundamental form. In general this is an extremely rare occurance and in fact an ordinary manifold does not contain any totally geodesic submanifolds other than geodesies themselves [1]. Totally geodesic submanifolds arise for example in Riemannian product manifolds if we consider 2- dimensional products of a horizontal geodesic and a vertical geodesic [1]. 1.1.2 Generalization of Simple Points As was previously mentioned, Cohn-Vossen along with Gromoll and Meyer were responsible for determining that a complete noncompact n-dimensional manifold of positive curvature is diffeomorphic to R " . In his proof of the 2-dimensional case, Cohn-Vossen demonstrated that simple points always exist in such manifolds. A point p in a manifold M is a simple point if there are no geodesic loops in M starting and ending at p or equivalently if {p}, the set consisting of the single point p, is totally convex [1]. Gro-moll and Meyer then showed by constructing compact convex sets in the manifold that simple points exist in all dimensions when K > 0. Similarly, Cheeger and Gromoll constructed compact totally convex sets in a manifold of nonnegative curvature to show the existence of a soul. Furthermore, the soul of a manifold also has the property of compactness and total convexity. Therefore, the existence of a soul in every complete noncompact manifold of nonnegative curvature provides a natural generalization of simple points 4 Chapter 1. Introduction in manifolds of positive curvature [4]. The existence of simple points and the presence and nature of a soul in a manifold allow one to draw strong conclusions as to the structure of the manifold. 5 Chapter 2 Comparison Theorems The behavior of geodesies in spaces of constant sectional curvature is well understood as we are equipped with formulas which allow explicit compu-tation of distances and angles. For an arbitrary Riemannian manifold that does not have constant sectional curvature, it is often possible to find a lower bound on the curvature. Once equipped with a lower bound on the sectional curvature, there are many theorems available which allow one to draw geometric conclusions based on comparison with a suitable space of constant curvature. Two comparison theorems in particular; namely the Rauch Theorem and Toponogov's Theorem, play a fundamental role in the proof of the Soul Theorem. These theorems are the focus of the remainder of this section. 2.1 Rauch Comparison Theorem The Rauch Comparison Theorem is a powerful tool used to understand the geometry of a Riemannian manifold M which satisfies K > S for some constant 5. Specifically, with some additional assumptions, the theorem de-scribes the relationship which exists between the magnitude of vectors in a Jacobi field along a geodesic in M with those in a particular Jacobi field along a geodesic in a suitable comparison space of constant curvature. We begin this section by providing the definition of a Jacobi field as well as some related properties. We will not provide proofs for the theory relating to Jacobi fields as they can be found in many standard books on the topic. We recommend the interested reader refer to [5] or [11] for a more complete treatment of the subject. Let 7 : [a,b] —> M be a geodesic and let F : (—£,£) x [a, b] —> M be a smooth variation of 7 such that F(s,t) = 7s(i) and F(0,t) = 70(t) = j(t). F is a variation through.geodesies if each 7S(£) is a geodesic. Let T = 7% = 7«(0 a n ( f let V = denote the variation field. A variation of 7 is illustrated in Figure 2.1. 6 Chapter 2. Comparison Theorems s F Figure 2.1: Smooth variation of 7 We are particularly interested in variations through geodesies because we want to know what effect curvature has on such a family of curves. With this knowledge comes a better understanding of the geometry of the manifold. If a family of curves is a variation through geodesies then it is possible to derive a second order ordinary differential equation which must be satisfied by the variation field. This equation, which we define in the next theorem is known as the Jacobi Equation. Theorem 2 .1.1. Let 7 be a geodesic and V a vector field along 7. // V is the variation field of a variation through geodesies, then V satisfies the Jacobi equation: VtVtV = R(T,V)T where V denotes the Levi-Civita connection and R(. ,.). is the Riemann curvature tensor on M. This theorem gives the following definition: Definition 2.1.2. A Jacobi field is any vector field along a geodesic which satisfies the Jacobi equation. A trivial example of a Jacobi field along a geodesic 7 : [0, /] —> M is the field of tangent vectors 7'(.s) for s £ [0, /]. An interesting property of Jacobi fields is the fact that the converse of the above theorem always holds. In particular, every Jacobi field along a 7 Chapter 2. Comparison Theorems geodesic 7 may be realized as a variation field for some variation of 7 through geodesies [11]. Now as one may suspect, since the Jacobi equation is a second order ordinary differential equation we may borrow from the theory of ordinary differential equations to obtain the following existence and uniqueness result for Jacobi fields. Theorem 2.1.3. Let 7 : [0,/] —> M be a geodesic, a e [0,1] and p = 7(a). Then for any vectors X, Y G TpM, there exists a unique Jacobi field J along 7 satisfying initial conditions J(a) = X and V tJ(a) = Y. The notion of a conjugate point is important in the theory of Jacobi fields and is required by the Rauch Theorem. In the definition below we state two ways to characterize conjugate points. Definition 2.1.4. (l)Let^ be a geodesic segment joining p,q € M. If there exists a Jacobi field along 7 vanishing at p and q but not identically zero then q is conjugate to p along 7. (2)Let p <S M, v € TpM, and q = exppv. Then q is conjugate to p along *y(t) = expp tv, t 6 [0,1] if and only if v is a singularity o/expp : TpM —» M. The second statement is the more intuitive one; conjugate points q = exppt) are the points where expp fails to be a local diffeomorphism at v. For example, let p be any point on the unit sphere in R 3 . The geodesies on the sphere are great circles and so there are infinitely many geodesies joining p with the antipodal point, say q. Therefore q is a singular point of expp and q is a conjugate point to p along any of these geodesies. This example illustrates another important property of conjugate points. The conjugate point on 7 which is closest to p is the point beyond which the geodesic fails to be minimizing. Although geodesies with interior conjugate points are not minimizing, it is possible to have non-minimizing geodesies which contain no conjugate points. For example geodesies on the cylinder in R 3 do not contain any conjugate points but once you travel more than halfway around a circular geodesic it is no longer minimizing [11]. An understanding of conjugate points is also useful in that it allows one to solve what is known as the "two-point boundary problem" for Jacobi fields. This problem requires one to find a Jacobi field J along 7 satisfying ./(a) = V and J(b) = W for any V € T 7 ( a ) M and W £ Ty{b)M. A unique 8 Chapter 2. Comparison Theorems solution exists if and only if 7(a) and 7(6) are not conjugate along 7. The Rauch Theorem also requires a generalization of conjugate points known as focal points. In order to define focal points we must consider a submanifold 5 of a Riemannian manifold M. For a given point p £ S, a vector n € TPM is said to be normal to S if for every v G TpS, (n, v) = 0. The set of all n 6 TpM which satisfy this property is called the normal space to S at p and is denoted by NPS or TpS-1. The normal bundle NS is a vector bundle over S defined by NS = Upgs NPS. This is similar to the tangent bundle TM to M at p which is defined as TM - \JpeMTpM. The normal bundle has a manifold structure and satisfies TM = TS0./VS. Conjugate points are the singular points of exp and a focal point q of S is defined to be a singular point of exp |jvs-The final definitions we require are those of a normal geodesic and a geodesic submanifold defined by a vector. Definition 2.1.5. A geodesic 7 : [a,b] —> M is a normal geodesic if I  i ll=i. Definition 2.1.6. For any p £ M, the geodesic submanifold defined by v € TpM is expp(U) where U is a neighborhood of the origin in {w £ TpM\w ± v} for which expp \ u is an embedding. We are now ready to state Rauch's Comparison Theorem. Following some initial assumptions, the theorem is divided into two parts; the first relates to conjugate points and the second concerns focal points. Theorem 2.1.7. (Rauch) Let M and Ma be Riemannian manifolds. Sup-pose dimM < dimMQ. Let 7 : [0, /] —> M and j0 : [0,1] —> Ma be normal geodesies. Assume K(o0) > K(o) for all plane sections oQ, a along 7 D and 7 containing 7^  and 7' respectively. (I)Assume further that for any t £ [0,1], "fo(t) is not conjugate to 7o(0) along j0. Let V and VQ be Jacobi fields along 7 and ja satisfying the follow-ing conditions (a) V(0) and Vo(0) are tangent to 7 and j0 (b) || V(0) || = || Vo(0) I  and \\ V'(0) || = || Vo'(0) || (c) (T(0), V'(0)> = (To(0), Vo'(0)> where T = 7', T0 = iQ Then for all t e [0,1], 9 Chapter 2. Comparison Theorems I  V(t) \\>\\v0{t) ||. (II) Assume that for any t G [0, Z], "f(t) is not a focal point of the geodesic submanifold NQ defined by TD. Let V and Va be Jacobi fields along 7 and "f0 satisfying the conditions (a) V(0) and Vo(0) are tangent to 7 and-f0 (b) || V(0) || = || Vo(0) I  and \\ V'(0) \\ = \\ Ko'(0) || (c) (T(0),V(0)) = (ro(0),V-o(0)> Then for all t G [0,/], I  V(t) \\>\\V0(t) ||. The following definition and lemma play a crucial role in the proof of the Rauch Theorem. Let 7 : [a, b] —> M be a geodesic. Def in i t i on 2.1.8. The index form I(V,W) given by W W) = Sa^rV, VTW) + (R(W,T)V,T) is a symmetric bilinear form on all smooth vector fields V and W along 7. It is interesting to note that the index form gives another characterization of Jacobi fields. If I is defined on all piecewise smooth vector fields which vanish at the endpoints of 7, then a smooth vector field V is a Jacobi field if and only if I(V, W) = 0 for all W [3]. L e m m a 2.1.9. Let 7 be a geodesic in M from p to q such that there are no points along 7 which are conjugate to p. Suppose W is a piecewise smooth vector field on 7 and V is the unique Jacobi field satisfying V\p = W\p = 0 and V\q = W\q. Then I(V,V) < I(W,W) with equality holding only in the case that V = W. Proof. Let 7 : [0,1] —> M be a geodesic from p to q. Let {Vi} be a basis of TqM. By the existence and uniqueness theorem for Jacobi fields we may extend each Vi to a unique Jacobi field along 7 satisfying Vi\p = 0. We may write Vi = t.Ai where t is the same parameter as in the definition of 7 and Ai is a vector field along 7. Since {V;} is a linearly independent set, {Ai} is also a linearly independent set. Since W is a piecewise smooth vector field 10 Chapter 2. Comparison Theorems along 7 there exist functions qi(t) such that W = YliQi(t)Ai- In fact, since W vanishes at p and Vi — tAi implies Ai = \Vi for all t not equal to zero, there exist functions fi(t) for which W = Ylifi^Wi- Since V is the unique Jacobi field satisfying V\p = 0 and V\q = W\q we have V = ^ fi{l)Vi. Using these definitions for W and V we will now determine expressions for /(V, V) and I{W,W). I(W)= f\v',V') + (R(T,V)T,V) Jo = / (V,vy- (v",v) + (R(T,V)T,V) Jo = / (V, V)' - (V", V) + (V", V) since V is a Jacobi field Jo (V,vy 0 <V'(1),V(1)> £>( i )v7( i ) ,£>( i ) i / t ( i ) ) i i We wish to show I(V, V) < I(W, W) so we now turn our attention to eval-uating I(W,W). By definiton I(W, W) = ^ {{W, W) + (R(T, W)T, W)) and using the defi-nition of W we expand the first term in the integral as follows f\w\w')= A(£/^)',(]T/^)'> Jo Jo i '(EiflVi + fiVJ), + jwi)) i i ' 0 i i i i ii Chapter 2. Comparison Theorems For simplicity we will evaluate the terms in the above integral separately. = T,[1fiMM,vj)'-M',vj)) id 0 = £( / i ( i ) / : , ( iXv;m^( i» J 0 The last step follows from using integration by parts to evaluate the first term in the preceding integral. Now the first term is I(V, V) so by making this substitution, expanding the second term, and using the Jacobi equation to simplify the last term we get A s E wi) = 1{y>v) - E [\f&M, v } > + v 3 ) ) - E f fifiWTMTtVj) i,j 0 = /(V, V ) - E fififiW, V3) + hf'jiVl, Vj)) W . E / i W E ^ - ) = /(v, v) - E fu'iliW, v3) + f.ftiv;, Vj)) - f (R(T,W)T,W) Jo =i(v,v)- A E ^ E ^ > - A E - ^ E - ^ - l\R{T,W)T,W) Jo i i Jo 12 Chapter 2. Comparison Theorems When we substitute this back into the expression for I(W, W), we obtain i(w,w) = i(v,v) - A E ^ ' - E ^ ) - A E ^ ' ' E ^ > Jo • j Jo l 3 - f (R(T, W)T, w) + f (E m, E fiV() Jo Jo i + A E / ^ E ^ > + A E / ^ ' E Z ^ ) Jo i i Jo i + [ (R(T,W)T,W) Jo The third and sixth terms cancel as well as the curvature terms and so we are left with nw, w) = i(v, v) - C(E E Wi) ' 0 i j + A E / # ' E ^ > + AE/^>E/^> Jo i i Jo i i If we can show (V(, Vj) = (V{, Vj) then we may cancel the second and third term and the above expression simplifies to I(W, W) = I(V, V) + ^ ( E fiV*> E fiVi Since the last term is nonnegative, we have I(V,V) < I(W,W). Equality holds when Yi f[Vi = 0. Since {V^} is a linearly independent set, Yi fiYi = 0 if and only if f-(t) = 0 for all i. Therefore, fi(t) is constant for every i, so in particular fi(t) = / i ( l ) for all t. Therefore, equality holds if and only if V = W as desired. Hence it remains to show (V/, Vj) = {Vi, Vj). Since «v7, Vj) - (vu vj)y = (v(', Vj) + (v;, vj) - (v(, vj) - (vt, vj') = (VlWj) - (VuVJ') = (R(T, Vi)T, Vj) - (V-, R(T, Vj)T) — 0 due to the symmetry of the curvature tensor then (V(,Vj) — (V.,Vj) is a constant. Since the expression vanishes when t = 0, the constant is zero, and therefore (V/, Vj) — (Vi, Vj). • 13 Chapter 2. Comparison Theorems Proof of the Rauch Theorem (I). For now we will assume that V, and V0, are perpendicular to T, and T0 or that || V(0) ||= (r(0),V'(0)) = 0 and || Vo(0) ||=<ro(0),V-o'(0))=0. Now consider the ratio |y where t is the same parameter used to define the geodesies 7 and 7G. As in the statement of the theorem, 7 G contains no points conjugate to 7o(0) along 7 Q so the ratio is defined for all t except t = 0. Using L'Hopital's rule we may evaluate the limit l i m i L | . l i m i ^ L = l i m | ir^ t_o I V0 ||2 t-o (V0, V0) t-,o 2(V0', V„) = lim <V">V) + <y''V') (212) These terms vanish at t=0 and so from the above calculation the limit simplifies to || v ||2 {V{o),v'{o)) = I no) I _ 1 in i Q\ ^ ° n m 2 (no),no)> 1 no) 1  1 j The last equality in the calculation holds by the assumption made in the statement of the theorem. . Now we wish to show that the ratio jj^ qp- is a monotone increasing function of t. In light of the fact that the ratio tends to one as t tends to zero the desired result || V \\>\\ V0 || will follow immediately. Since v II2\ _ (V0,Vo)i(V,V)-(V,V)i(V0,Vo) > 0 dt\\\V0fJ (i/ 0 ,V 0) 2 if and only if 2{V0,V0)(V',V) - 2(V,V)(V^,V0) > 0 if and only if if and only if (V0,V0)(V\V)>(V,V)(V:,V0) (V',V) > (VLV0) (V,V) - (V0,V0) we will show is increasing by showing ^'yj > ^ o r a u ^ > 0-14 Chapter 2. Comparison Theorems Let s e [0,1) and consider the vector fields which are constant multiples of V and VQ defined by By the construction of Ws(t) it is clear that (V, V) _ (Ws, W's) (VI, V0) _ (W'0s, W0s) s,riQ (2.1.4) Since || W,(s) ||= left hand side becomes (V, V'\ (V,V) (WS,WS) (V0,V0) (W0a,W0s) 1, when evaluated at t = s, the relation on the (V,V) I  Wa{s) || [S(ws,w's Jo f Jo f Jo f (W'S,W'S) - K(o) || Ws Jo (w>,w's) + (w';,ws) - (R(WS,T)T,WS) (2.1.5) (2.1.6) (2.1.7) (2.1.8) (2.1.9) (2.1.10) where er is the plane section spanned by T and > a n < ^ (2-1-9) follows since Ws is a constant multiple of a Jacobi field. Let P-1 denote the parallel translation along 7 in the direction from q to p and let Plo be the parallel translation along j0. Define I : Ty^M —> T7 o(0)(Mo) so that I is an injection which preserves the inner product and It °: T 7 ( t ) M -» Tlo(t)(M0) by It(X) = Plo o / o P_ 7(X). In addition, we will assume h(T) = T0 and /S(W5) = W 0 s . Finally, define a field W 0 s by W^a(t) = h(Ws(t)) Since 7 preserves the inner product (Ws(t),Ws(t)) and (Wfr),Wa(t)) 15 = (W,,.(t),Wo.(0)) Chapter 2. Comparison Theorems Given these new structures and since we know K(o0) > K{o) we may pro-ceed with the calculation given in (2.1.10) (VX) (W) Jo i: (W's, W's) - K{o) || Ws f (W>a,W>s) - K(o) || WQs > / (W03,W>S)-K(CT0)\\W03 ||2 Jo = [\w^s,W^J - (R0(W7,,T0)T0,W;S) Jo > I{W03,Wos) by Lemma 2.1.9 = /V;,<>- (R(W0s,T)T,W0s) so we have (V,V) (V,V) > I*'«,,<,)- {Ro(W0s,T)T,W0s) Jo = /Vo,.W^.>-«.^o.> Jo = fs(w0„w0,y Jo = «sXoa)\s _ (VoX) (Vo, V0) using (2.1.4) (2.1.11) (2.1.12) (2.1.13) (2.1.14) (2.1.15) (2.1.16) (2.1.17) Since this holds for any 5 € [0,1], we have > (vTy') for all t as desired. We have shown the theorem holds for the specific case when V and VQ are perpendicular to T and TQ or when both (T(0), V"(0)) and (To(0), Vo'(0)) equal zero. Now we must consider the general case. Decompose V and VQ as follows: V = V+(T,V)T and V0 = V0 + (T0, V0)T0 16 Chapter 2. Comparison Theorems where V and V0 satisfy the conditions of the first case. Therefore, || V(t) \\>\\ V0\t) || holds for all t. Now since T(T, V) = ( V T T , V') + (T, VTV) = (T,V") = — (T, R(V, T)T)by the Jacobi equation =-(R(T,T)V,T) = 0 by definiton of the curvature tensor (T, V) is constant on 7. Therefore (T, V) = (T(0), V'(0)), or in other words T(V,T) = (T(0), V'(0)). This implies that (T, V) = (T(0), V"(0))i+C where C is a constant. Evaluating at t = 0 we have C = (r(0), V(0)) and hence (T, V) = (T(0), V'(0))t + (T(0), V(0)) (2.1.18) The same argument may be used to show {T0, V0) = (T o(0), Vo'(0))t + (T o(0), Vo(0)> (2.1.19) By assumption (T(0), V"(0)> = (T o(0), Vo'(0)> and (T(0), V(0)) = (T o(0), Vo(0)> therefore we have (T,V) = (T0,V0) from (2.1.18) and (2.1.19). Hence from the decomposition we may now conclude || V(t) \\>\\V0(t) [[ for all £ e [0,Z]. • The proof of part (II) of the Rauch Theorem requires a slightly modified index lemma but the techniques involved are identical to those used in the proof of part (I). Therefore, we will not present the details here but instead we recommend the reader refer to [3]. Several useful corollaries of the Rauch Theorem have been established and can be found in [3] or [16]. One particular corollary is necessary in the proof of the Soul Theorem and so we state it below. Corollary 2.1.10. (of Rauch II) Let 7 : [0, /] —> M , and 7 : [0,1] -» M be normal geodesies and let E and E be parallel unit vector fields along 7 and 7 which satisfy (E(t),f(t)} = (E(t), ^ '(t)). Let Nt denote the hypersurface 17 Chapter 2. Comparison Theorems exp7((){u; € T7( t)M|u; _L E(t), \\ w \\< e,e > 0 is small} Nt is perpendicular to E(t) and totally geodesic at 7(t). Let f(t) : [0,1] —> K be a C°° function such that f(t) is not greater than the first focal value fo{Nt) of Nt along a geodesic s i—> exp7(() sE(t). Let c : [0,/] —> M be a smooth curve defined by c(t) = ex.pf(t)E(t) and let c : [0, /] —> M be defined by c(t)=expf(t)E(t). Assume that Ka > K^, for all plane sections a, a in M and M respectively. Then L[c\ < L[c] where L[c] denotes the length of the curve c. Remark 2 ,1.1. It is useful to note that in the case where L[c] = L\c] the curves c(t) = exp7(t) f(t)E(t) determine a flat totally geodesic rectangle in M. 2.2 Toponogov's Theorem Toponogov's Theorem is a global generalization of the first Rauch Compar-ison Theorem. This useful theorem provides information about distances in a manifold with sectional cuvrvature bounded below relative to distances in a suitable comparison manifold of constant sectional curvature. Some basic definitions necessary for the statement of Toponogov's The-orem are as follows: Definition 2 .2.1. A geodesic triangle (71,72)73) in a, Riemannian man-ifold M is a figure consisting of three distinct points p\, p2, P3 called vertices and three geodesic segments 7, parametrized by arclength, joining pi to Pi+\ for i — l,2,3(mod 3) such that li + li+\ > ^ +2 where li denotes the length of 71 and ^ = (^7^ (0),—7^ 2(^ +2)), the angle between the tangent vectors of the geodesic segments meeting at pi. Figure 2.2 depicts a geodesic triangle according to this definition. 18 Chapter 2. Comparison Theorems Figure 2.2: A geodesic triangle Definition 2.2.2. A geodesic hinge (71,72,0:) is a figure in a Riemannian manifold M consisting of geodesic segments 71,72 such that 71 (Zi) = 72(0) anda = Z(-7 , 1(i 1),y 2(0)). Figure 2.3 illustrates a geodesic hinge. Figure 2.3: A geodesic hinge The first statement of Toponogov's Theorem concerns geodesic triangles while the second statement concerns geodesic hinges. These statements will be referred to as A and B respectively, and although the details are omitted here it is possible to show equivalence between them. Theorem 2.2.3. (Toponogov's Comparison Theorem) Let M be a complete manifold with sectional curvature K > 5 for some constant 5. Let Ms be the simply connected 2-dimensional space of constant curvature 5. 19 Chapter 2. Comparison Theorems (A) Let (71,72,73) determine a geodesic triangle in M. Suppose 71 and 73 are minimal and if 5 > 0 suppose I2 = £[72] < Then there exists a geodesic triangle (71,72,73) in Ms with the same side lengths L[7i] = £[7i] /or i = 1,2,3 and o7i < d\ , 0 3 < a 3 . 77ie triangle in Ms is uniquely deter-mined except when both 5 > 0 and L[7i] = =^ hold for some i. (B) Let (7i,72,ct) determine a geodesic hinge in M. Suppose 71 is min-imal, and if 5 > 0 suppose also that I2 = £[72] < £e£ (71,72, ct) be the geodesic hinge in Ms satisfying L[7i] = £[7»] for i = 1,2. Then d(7i(0),72(/2)) < d(7i(0),72(^2)) This theorem was first established by Toponogov but for a simpler proof of this result we refer the reader to a more recent argument by Karcher [10]. Toponogov's Comparison Theorem has become one of the most useful theorems in Riemannian Geometry. There are many applications of the theorem in addition to the fundamental role it plays in the proof of the Soul Theorem; one.of the most celebrated of which is its usefulness in the proof of the Sphere Theorem. The Sphere Theorem states that if a complete simply connected n-dimensional manifold has sectional curvature satisfying I < K < 1, then M is homeomorphic to a sphere [1]. Manifolds satisfying such a curvature condition are said to be quarter pinched. A weaker version of this theorem which required the sectional curvature to satisfy | < K < 1 was proven by Rauch in 1951 but the establishment of the Sphere Theorem was attributed to Klingenberg and Berger in the early 1960's. After this finding, mathematicians began to work on a stronger result of determining the pinching constant a manifold must satisfy in order to conclude the man-ifold is not only homeomorphic but diffeomorphic to a sphere. This long standing open problem was known as The Differentiable Sphere Theorem. In an exciting paper submitted in May 2007, S. Brendle and R. Schoen pro-vide a proof of the quarter pinched Differentiable Sphere Theorem [2]. The techniques used in the proof focus on the Ricci flow and are very different from those discussed in this paper. 20 Chapter 3 Preliminaries for the Proof of The Soul Theorem Totally convex sets play a fundamental role in the proof of the Soul Theo-rem and so we begin this section by defining the various notions of convexity which will be required. For the following definitions assume C is a subset of a Riemannian man-ifold M . D e f i n i t i o n 3 .0 .4 . A set C is totally convex if whenever p, q € C and 7 is a geodesic segment from p to q, then 7 C C . D e f i n i t i o n 3 .0 .5 . A set C is strongly convex if for any points p, q € C there exists a unique normal minimal geodesic 7 joining p to q, and 7 C C. D e f i n i t i o n 3 .0 .6 . If for any point p e C, the closure of C, there exists an e(p) > 0 such that C f | B t ( p j ( p ) is strongly convex, then C is said to be locally convex or simply convex. R e m a r k 3 . 0 . 1 . (1) A totally convex set is convex and connected [3]. (2) Any nonempty intersection of finitely many locally convex set is locally convex. The same property holds for strongly convex sets as well [16]. (3) The closure of a convex set is convex [3]. (4) For any p S M, there exists a continuous function r : M —> [0, 00) such that any metric ball contained in Br^(p) is strongly convex. This function is referred to as the convexity radius at p. In the definition of local convexity, e(p) < r(p). For a proof of the existence of the convexity radius we refer the reader to [16]. An interesting property of totally convex sets which is also a critical observation in the proof of the soul theorem is contained in the following theorem. The proof of this statement was originally presented in [4]. The details presented here are based on arguments found in [3], [4], and [16]. 21 Chapter 3. Preliminaries for the Proof of The Soul Theorem Theorem 3.0.7. Any closed totally convex set C contains a set N (possibly with dN ^ 0 ) which has the structure of a smooth imbedded totally geodesic submanifold. Proof. Let C be a totally convex set in M. By the definitions above, C is connected and locally convex. Let k < dim M be the maximal dimension of all imbedded submanifolds of M that are contained in C. Let {Na}aGA be the collection of all such fc-dimensional submanifolds and take N = \Ja Na. We begin by showing N is an imbedded submanifold of M. Let p £ JV. We must show there exists an open neighbourhood U of p in N and a diffeomorphism ip : U —> Rfc. Now since p £ N, we must have p £ Na for some a. We know Na is a submanifold of M , so there exists a coordinate neighbourhood V of p in Na and a diffeomorphism ip : V —> R fc. Without loss of generality we may choose V C B^. It suffices to show that the 2 chart (V,ip) on Na is also a chart on N. For this to hold, we must show that V is an open set in N which has the induced topology from M. Consider the set Ns(V) = {v€TqN± :\\v\\<6,qe V} where 0 < S < is chosen so that the normal exponential map exp-1 : N$(V) —> M is a diffeomorphism onto its image. By the inverse mapping theorem, since Ng(V) is an open set in the nor-mal bundle of Na, and exp-1 is continuous, the image T$(V) = exp±(N/i(V)) is an open subset of M. The open sets of N are those of the form 0 n N where 8 is an open set of M. Therefore, by the above argument Ts(V)nN is an open set in N. It now suffices to show V = Ts(V) fl N since if this holds then (V, ip) is a chart on N. We will proceed by contradiction and assume V ^TS(V)DN. Since V / T5(V) n N, there exists a point q € N satisfying q € TS(V) n N but g ^ V. By definition, since q € N we have g £ Np for some /3. Let r £ ATQ be the point closest to q. Since Na and are submanifolds of M , the minimal geodesic 7 joining g and r is orthogonal to V. Then since Na is a smooth submanifold and exp,j is a diffeomorphism from a neighborhood of the origin in TqM to a neighborhood of r in M , there is an open neigh-borhood V C V of r in Na, such that the minimal geodesies from q to the points in V' meet V transversally. 22 Chapter 3. Preliminaries for the Proof of The Soul Theorem N0 Figure 3.1: Geodesic Cone We claim that the cone defined by A = {expqtv\v e TqM, \\v\\ < e(q),expqv e V',0 < t < 1} formed by this subset of geodesies is a (k + 1)—dimensional submanifold of M which is contained in C. The cone is illustrated in Figure 3.1. Let H = {tve TqM\exprv£ V, \\v\\ < e{q),0 < t < 1} V is an open subset of a fc-dimensional manifold, so e x p ~ 1 ( V ) is a k-dimensional submanifold of the tangent space TqM, and the rays from the origin in TqM meet exp^ 1 (V / ) transversally. Therefore the cone H is a (k + l)-dimensional submanifold of TqM, and so A = expqH is a (k + 1)-dimensional submanifold of M . Since C is totally convex and the cone is formed by connecting points in N C C with geodesies, the cone is contained in C. Hence, there exists a (k + l)-dimensional submanifold of M contained in C. This contradicts the minimality of k. Therefore N is an imbedded submanifold of M. Now we must show that N is totally geodesic. Let 7 be a geodesic be-ginning at p which is contained in N. Suppose also that 7 is not a geodesic of M. Let (. < e(p) be such that 7][o,e] is a minimal geodesic from p to 23 Chapter 3. Preliminaries for the Proof of The Soul Theorem q = -y(e). Since M is complete there exists a minimal geodesic 7 in M be-tween p and q. If 7 were not contained in N then using arguments as above we would obtain a contradiction. From the discussion concerning geodesic submanifolds in the introduction it is clear that every geodesic of M which is contained in iV is a geodesic of AT. Therefore 7 and 7 are two distinct minimal geodesies of N which share the same endpoints. But since p, q, are contained in Be(p), the minimal geodesic between these points is unique. Therefore, -y| [0, e] = 7. Hence, 7 is a geodesic of M. Therefore by definition, N is totally geodesic. • The totally geodesic submanifold which arises from this theorem is not necessarily the soul of the manifold. The remaining properties we require of the soul is that it is compact and without boundary. Since some of the details are quite technical we include here a brief overview of how we intend to show the existence of the soul in a manifold. First we must show that the interaction of completeness and nonnega-tive curvature allows one to easily form a totally convex set associated to any point in the manifold. From this totally convex set it is then possible to construct a family of compact totally convex sets. In the next chapter, we will formulate a theorem; referred to by Cheeger and Ebin as the key to constructing the soul, which provides a way to make these sets as small as possible while still retaining the properties of compactness and convexity. By iterating this theorem one obtains a compact totally convex set without boundary and it is by applying the previous theorem to this final set that one arrives at the soul of the manifold. We begin by recalling two basic definitions from Riemannian Geometry: Definition 3.0.8. A Riemannian manifold is complete if for all p £ M, the exponential map, expp is defined for all v £ TpM, ie. if any geodesic 7(i) starting at p is defined for all values of the parameter t 6 R Definition 3.0.9. A ray 7 in a Riemannian manifold M is a geodesic 7 : [0, 00) —> M parametrized by arc length which minimizes the distance between 7(0) and f{s), for any s £ (0, 00). Every point p in a complete noncompact manifold M is the starting point of a ray, contained in M. Consider the following argument: Let p £ M . Since M is noncompact it is possible to choose a sequence of points {pi}ieN such that lim^oo d(p,pi) = 00. Since M is complete there exists a sequence 24 Chapter 3. Preliminaries for the Proof of The Soul Theorem of normal geodesies {7i}igN joining p and pi such that the length of 7* is the minimal distance between p and pi (by the Hopf-Rinow theorem). By compactness of the unit sphere in TpM there exists an accumulation point v for the sequence of vectors (7,'(0)}. The existence of a ray 7 starting at p fol-lows by defining 7 : [0, 00) —» M to be the geodesic with 7(0) = p and 7' = v. Since the requirement of completeness necessitates the existence of rays, the topology of a complete manifold of nonnegative curvature is greatly re-stricted in order to satisfy this property. In fact, the existence of a ray associated to every point in the manifold allowed Cheeger and Gromoll to associate to each point in the manifold, a totally convex set. The construc-tion of such a set is quite ingenious and involves Toponogov's Theorem. The details are described in the following theorem. Theorem 3.0.10. Let M be a complete noncompact manifold of nonneg-ative sectional curvature. Then M contains at least one totally convex set associated to each p G M. Proof. Choose any p G M. By the previous argument there exists a ray 7 : [0, 00) —> M starting at p. For every t, we may form a metric ball Bt(j(t)) with center j(t) and radius t. Now consider the union i? 7 = (Jt>o Bt(l(t)) of these metric balls along 7. The desired totally convex set emerges when we take the closed complement of this union. For this, we define H7 := M\B7 as depicted in Figure 3.2. It is also important to note that p G since Bt(l(t)) being open for every t implies p £ Bt(j(t)). We will now show, using a proof by contradiction, that for any ray 7, 7/7 is a totally convex set associated with p. To begin, assume i / 7 is not totally convex. This implies that there exists a geodesic 7 0 : [0,1] —> M with 7o(0), and 7o(l) G H-y, but f0(s) 0 H-y for some s G (0,1). For ease of notation let q = 7o(s). Since q g" Hy, then in particular q G Bt("i(t)) for some t > 0. In fact, there exists a t0 > 0 such that q G Bt{~i(t)) for all t > t0. This is true since by the construction of the metric balls along 7, i?t2 (7(^ 2)) D Bt1(j(ti)) for all t\ < £2-Now let d(q,j(t0)) — tD — e for some e > 0. Then d(q^(t)) < d(g,7(*o)) + d(7(to).7(*)) = t0 - e + t - t0 = t-(. 25 Chapter 3. Preliminaries for the Proof of The Soul Theorem Figure 3.2: Construction of Hj. for all t > tQ since 7 is a ray. For each t, let 7o(st) be a point on jQ which is closest to ~f(t). Consider the minimal geodesies 7o = 7o|[o,st] •y\ from jo(st) to >y(t) 7^  from 7(i) to 7o(0) Since q is a point on 7 G and 70(.s't) is the closest point on 7 0 to j(t) we know L[y{] < d(q,-y(t)) = t - e. Or in other words t > L[y{] + e. Also since 7o(0) G E1 then in particular 7„(0) 0 Bt{"i{t)) and so d(7o(0), 7(f)) > Therefore £[73] > d(7o(0), 7(0) ^ By definition 7J and 73 are minimal geodesies and since £[7*] < £[7o] is finite we have + L\y\] > Z/[70] f° r sufficiently large Therefore (7^, 71,72) forms a geodesic triangle in M. Since M has nonnegative curvature, Toponogov's Theorem (A) implies that there exists a triangle (7*,71,72) in euclidean space with Z/py*] = Z/py,-] for all i = 1,2,3 and a\ < a\ for i = 1,2. Here the angle a* corresponds to the angle opposite the side 7'. But we also know that a2 = § since M is complete and j0(st) is the closest point on f0 to the point 7(4) on M . 26 Chapter 3. Preliminaries for the Proof of The Soul Theorem Therefore, Toponogov's theorem implies that in particular a2 < f • If we consider the geodesic triangle in the euclidean plane, we may use the law of cosines as follows: L2m = L2m + L 2 [ 7 l ] - L2[^]L2[y{} cosa\ But a | < f implies that coso^ > 0 and so the last term in the above expression is positive. Therefore, Now recall L ^ ] > t and L[j[] —t — t. Using Toponogov's Theorem we may substitute these values into the above inequality to obtain t2 < L2tf2] < L2[lo] + (t - e)2 or simply t2<L2h0} + (t-e)2 which expands and simplifies to t2 < L2[lo] + t2 - 2et + e2 if and only if 2te < L2Yi0] + e2 or t^L2[lo] + e2 e But this is impossible for sufficiently large t since L[y0} is fixed. Therefore we have a contradiction, and hence Hy is totally convex. • Now that we have established the existence of a totally convex set asso-ciated to each p € M , the next step is to show that it is actually possible to associate to each p £ M a family of totally convex sets carrying the addi-tional property of compactness. We present below a proposition of Cheeger and Gromoll which proves the existence and some properties of such a family of sets [4]. Proposition 3.0.11. With M as above andp £ M, there exists a family of compact totally convex sets Ct, t > 0, such that: (1) t2 > t\ implies Ct2 D Ctx and Ct, = {qeCtM<l,dCt2)>t2-h} 27 Chapter 3. Preliminaries for the Proof of The Soul Theorem in particular, dCtx = {q € Ct2\d(q,dCt2) = t2-tl}. (8)\Jt>0Ct = M, (3) PedC0. Proof. Let p € M and let 7 : [0, 00) —> M be a ray starting at p. Let 7t : [0, 00) —> M satisfying 7 t(s) = 7 ( s + i ) denote the ray restricted from 7(4) to 00. We will show that the family of sets {Ct}t>o defined by Ct — f ] 7 H^t where the intersection is taken over all rays 7 starting at p, satisfies the requirements of the proposition. By the previous theorem each Hj is totally convex and closed. There-fore, for each t, Ct is totally convex and closed according to Remark 3.0.1. To show Ct is compact we will use a proof by contradiction. If for some t, Ct is noncompact then there exists a sequence of points {pi}^i in Ct satis-fying l i m j _ 0 0 d(p, pi) = 00 Let 7, : [0,/?j] —» M be minimal normal geodesies from p to pi. By definition, p € Hlt for every ray 7, so p £ Ct- There-fore the total convexity of Ct implies that every 7; is contained in Ct. By compactness of the unit sphere in TpM, there exists an accumulation vector v £ TpM of the set {7^(0)} for which a subsequence of {7;} converges to a ray 7 „ : [0, 00) —• Ct. Thus we have found a ray 7 v emanating from p which is entirely contained in Ct- This is impossible by the definition of Ct since in particular 7,, <£ . Therefore, Ct is compact. Now assume t2 >t\. We have Bs(lt2(s)) = BsMs + ti)) C Bs+t2-h(l(s + t2)) = B 8 +t a - t , (7t l (s + i 2 - t i ) ) in particular Bsht2(s)) C Bs + t 2_ t l(7 t l(s + t2 - h)) implies that \jBs(lt2(s))c\jBs(ltAs)) s s or by the notation introduced in the previous proof B~?t2 C Byii 28 Chapter 3. Preliminaries for the Proof of The Soul Theorem for any ray. Taking complements of these sets gives Hii2 = (Bi,.2)c 3 (5 7 ( l ) c = Hlti and thus Ct 2 D Ctl To prove the remainder of part (1) we require Ctl ={q€Ct2\d(q,dCti)>t2-t1} We claim that it is enough to show B1H= {q\d{q,Bll2) <t2-h} for any 7 emanating from p. If this were true then we would have {JB1H ={q\d(q,\jBli2)<t2-h} 7 7 where 7 ranges over all rays emanating from p. Hence Ctl = p| Hlti = p|(S 7 l i )c = (|J Byti )c = {q\d(q, [j Blt2 )>t2-t1} 1 1 1 1 (3.0.1) Now if we substitute d(q,\jB^) = d(q,d^[JB^) = d(g,5^P|<)) = d(q,8Ct2) into the final expression in (3.0.1) we obtain = {q\d(q,DCt2) > t2 — £1} as desired. Now it only remains to prove the claim that Blti = {q\d(q, Bll2) < t2 - ti}. 29 Chapter 3. Preliminaries for the Proof of The Soul Theorem First, let q be such that d(q,Byi ) < t2 — t\. Then there must be a point q' € Byt2 such that d(q.q') < t2 — <i- Therefore when s > 0 is sufficiently large, q' G Bs(jt2(s))- Hence q G B s +t2_ t l(7t2(s)) by the triangle inequal-ity. But we previously showed that B 7 ( 2 C Byi , therefore q G B 7 ( . Thus {q\d{q,Byi2) < t2 - t i } C B1H. On the other hand, if q G Blti then for some s > 0, q G B s + t 2 _ t l ( 7 t l ( . s + i 2 ~h)) = B s + t 2_ t l( 7 t 2(.s)) D B s ( 7 t a ( « ) ) Therefore, d(g,J3 7 t 2) <d(<7,B a(7t2(s))) < * 2 - t i as shown in Figure 3.3. Thus Blti C {q\d(q, B 7 , 2 ) < t2 — t\}. B a +t a - t 1 (7t 1 (s + *2 -«i)) Figure 3.3: Proof that d(<j, B 7 ( 2 ) < t2 - ti This establishes part (1) of the proposition. To prove part (2), take any q G M and let t > d(p,q). Then for any A' > 0 and any ray 7 emanating from p, the triangle inequality implies that d(q, Bs(lt(s))) + d(p, q) > d(p, Bs{lt{s))) = t. 30 Chapter 3. Preliminaries for the Proof of The Soul Theorem But d(p, q) < t implies that d{q, Bs(-yt(s))) > 0. Therefore g G (£?s(7t(s)))c = Hyt. Since this holds for all rays, q € Ct- Therefore, \Jt Ct = M. Part (3) of the proposition follows immediately from the definition of C0. • The proof of the above proposition follows closely the original proof pre-sented by Cheeger and Gromoll in [4]. An alternative approach is presented separately by P. Peterson [14] and T. Sakai [16]. These proofs make ex-plicit use of a map called the Busemann function. For any ray 7 in M , the Busemann function 67 is defined by B^iq) = l im t _ t + 0 O (i - d(q,-y(t))) The Busemann function is defined on all of M since there exists a ray as-sociated to every point in a complete manifold. Peterson and Sakai used the convexity of the Busemann function and defined the family of compact totally convex sets by Ct = fl7{^ 7 1 ( ( ~ 0 0 > *])} t o P r ° v e the proposition. These sets are the same as those in the family which Cheeger and Gromoll defined without explicit mention of the Busemann function. We refer the reader to [14] and [16] for a detailed proof using the Busemann function. Before we explicitly construct the soul of the manifold, it would be useful to know more about the set N C C from Theorem 3.0.7 which has the structure of a smooth imbedded totally geodesic submanifold. Recall, by definition N = \Ja Na where {Na}aeA is the collection of all imbedded submanifolds of M of maximal dimension. We will see in Lemma 3.0.13 below that N is connected and C is contained in the closure of N but first we require the following technical lemma. L e m m a 3.0.12. Let C be convex and connected, and let peCnN, p ' 6 B i £ ( p ) ( p ) n C , qe Bit(p)(p)nN Let 7 be the normal geodesic in M such that 7|[o,e] * s the minimal segment from q to p' where e. = diq,p'). Then 7|[o,e) C Af and hence p' £ N. If furthermore p' £ N, then f{s) £ C for all e < s < e + |e(p). Proof. Let 0 < e < e + \e(p) and p = 7(8) be a point in C. Since q, p' are elements of B^ip) and e = diq,p') then we must have e < 2 {^^j = f^-^ -Also since p = 7(e) e C where e < t + \t{p) < ^ + ^ < (f) e(p) then 31 Chapter 3. Preliminaries for the Proof of The Soul Theorem P e B e(p) ( p) as well. Hence p G C f | B e ( P ) ( p ) . Let W be a (k — l)-dimensional hypersurface in (p) f] N which passes 4 through q and is transversal to 7 at q. Now consider a geodesic cone V = {exp(iu;)|u; G TpM, \\w\\ < e(p), exp(w) G W, 0 < t < 1} as depicted in Figure 3.4. Figure 3.4: Hypersurface and corresponding geodesic cone The cone consists of geodesies from p to points in W. Since both p and W are contained in B e( p)(p) f]C, the strong convexity of the set implies there exists a unique minimal geodesic contained in Bcip^{p)C\C between p and points in W. Hence the geodesies forming the cone must be contained in C. By the same argument presented in Theorem 3.0.7, the geodesic cone is a smooth k- dimensional submanifold of M . But since N = ( J a Na where each Na is a k—dimensional submanifold of M which is contained in C; this cone must be one of the ATQ's. Therefore, V C N. Then in particular if we let p = p', we have 7|[o,c) C N since this geodesic segment is necessarily part of the cone. Hence p' G N. The final statement of the theorem follows from the observation that if e < (. < e. + ^ then p = 7(e) G C is farther along the geodesic than p'. Hence p' must be contained within V C N. Therefore if p' g N then 7(e) £ C for any e < e < | as desired. • 32 Chapter 3. Preliminaries for the Proof of The Soul Theorem Lemma 3.0.13. N is connected and C C N. Proof. Begin by considering the partition of N into maximal nonempty con-nected subspaces. Let the partition be denoted by {Nl}i^j where I is an index set. The strategy of the proof is to show that for any connected com-ponent Nl in the partition, C C Nl. We already have by construction that Nl C TV C C so if we can show C <Z N% then we have ./V* dense in C. Since this holds for any i 6 I, there must only be one unique component in the partition, namely N. Therefore, we may conclude C C N. It remains to show C C JVl. The result follows from a proof by contradiction. Suppose C <£ N\ Since C <£ N\ there exists a point p' e (C\ Ni). Since C is connected we may choose p 6 Nl n C such that p' 6 (p) CI (C \ Nl) 4 and q € B^(p) fl Nz. But these points satisfy the assumptions of Lemma 4 3.0.12, therefore p' € Nl. This is a contradiction, hence C C Nl • 33 Chapter 4 Proof of The Soul Theorem 4.1 Constructing the Soul In order to construct the soul of the manifold we must begin with a totally convex set which is as small as possible. We know that every complete non-compact manifold of nonnegative curvature contains compact totally convex sets. To prove the existence of the soul we require a way to contract these sets while allowing them to retain the properties of compactness and con-vexity. Cheeger and Gromoll [4] discovered an efficient way to accomplish this task. They began with a compact totally convex set C with nonempty boundary and found that the set of points at a maximal distance from the boundary of C is itself a totally convex set. They proved that the new set is smaller in the sense that the dimension is less than that of the original set and so this contraction procedure need only be applied a finite number of times before arriving at a set which is as small as possible, namely one with empty boundary. This is the idea behind the following theorem which we will now state formally. T h e o r e m 4.1.1. Let M have nonnegative curvature and let C be a closed and totally convex set with dC ^  0. Define the sets Ca, and Cmax as Ca = {p& C\d{p, dC) > a}, Cmax = p| Ca Then (1) for any a, Ca is totally convex, (2) dim C m a x < dim C The proof of this theorem requires that we first introduce the notion of a tangent cone and establish two lemmas. For the remainder of this section, let N denote the totally geodesic submanifold which exists by Theorem 3.0.7, and satisfies N C C C N by Lemma 3.0.13. Def in i t i on 4.1.2. The tangent cone Cp at p g C is the set 34 Chapter 4. Proof of The Soul Theorem {v £ TPM\exp(j^jj) £ N for some positive t < e(p)} \J{op} where op denotes the origin in the tangent space TpM. Let Cp denote the subspace of TPM generated by Cp. Lemma 4.1.3. Let C C M be a closed locally convex set. Suppose that there exist p £ dC, q £ intC, and a minimal geodesic 7 : [0,1] —> C from q to p such that L[j] = d(q, dC). Then Cp \ {op} is the open half-space H = {veCp\{op}\Z(v, - 7 ' (0 )<f} . Proof. Let 7 : [0,1} —* C be a normal geodesic joining p £ dC to a point q £ N such that the length of 7 realizes the distance between q and the boundary of C. We will first show that H C Cp\{op}. Let v £ H and choose s £ (0,/) so that d(p,7(s)) < Since 7 is a minimal geodesic, the dis-tance between 7(s) and dC is / - s. Therefore B;_s(7(s)) ndC = {p}. Since 7 is orthogonal to the boundary of C and v € H means Z(v, — 7'(/)) < §, we have expp contained in the interior of C for some t £ (0, e(p)). Hence, HcCp\{op}. Now to show Cp\{op} C H we will first use a proof by contradiction to show that Cp \ {op} c H. Suppose v £ Cp \ {op} but v £ H. By definition v 0 H implies Z(w, ~7'(0) > f • But this holds if and only if Z(-v,-j'(l)) < f. In other words, we have —v £ H. By the previous argument, every ele-ment of H is an element of Cp \ {op}, so in particular — v £ Cp \ {op}. By definition, this implies expp is contained in JV for some t £ (0,e(p)). Therefore, when e is sufficiently small, there exists a geodesic starting at p with initial direction —v denoted by 7_w for which "f-v(e) is contained in ./V. Equivalently, 7v(—e) is an element of N. Now consider 7v|[_C)o], a minimal geodesic joining a point 7„(-e) in AT to p. By Lemma 3.0.12 with q = 7„(-e) and p' = p, we have 7«(s) ^ C for all 0 < s < ^  since p' is in dC and hence not in N. Therefore, jv is not contained in A^  beyond p. But this is a contradiction since v £ Cp implies 7„(t) is contained in N for some £ £ (0, e(p)). Hence C p \ {op} C H. To complete the proof it suffices to show Cp \ {op} is an open set in Cp. Without loss of generality, assume q £ (p) n N. Recall 7 is a minimal 4 geodesic from p to q and let —7 denote the same geodesic but parametrized from q to p. Let 7 have length e = d(p,q). Then by Lemma 3.0.12, since p ^ N, -f(s) g" C for e < s < e + Therefore, -7 is not contained in C 35 Chapter 4. Proof of The Soul Theorem beyond p. This implies (-7)'(e) g" Cp but -(- 7)'(e) = T'(0) G C p . Hence Let v £ Cp\ {op} and consider the curve c(s) = exppst>. By definition and Lemma 3.0.12, c(s) is contained in N for all sufficiently small s > 0. Let crs denote the minimal geodesic joining the point c(s) to q. Let P^ denote the parallel translation along <JS from c(s) to Then since N is totally geodesic, Paa{c'{s)) is contained in TQN. Since v = lims_*o c'(s) it follows that P C T » = P » = lims^o Pas(c'(s)) is contained in TQN. Since this holds for any v £ Cp\{op}, we have P 7 ( C P ) C TQN which implies C P C P-Y(TQN) or in general C P C P_7(T,iV). Figure 4.1: Parallel Translation To see that Cp \ {op} is open in P_7(TgA^), consider an open neighborhood A c^(5) of c(.s) contained in ./V and minimal geodesies from p to every r £ Nc/Sy By the above argument, and since A ,^,^ ) is open in TV, the set of parallel translations along 7 of the associated initial tangent vectors of these geodesies forms an open neighborhood of P7(i») in TqN. Therefore the set of initial tangent vectors forms an open neighborhood of v in P_ 7(T 9A r). Each of these initial tangent vectors is, by definition, contained in Cp\{op}. Thus, using this construction, for every v £ Cp \ {op} there exists an open neighborhood of v in P_ 7(T ?A r) that is contained in Cp\{op}. Hence C p \{o p } is an open subset in Cp. • 36 Chapter 4. Proof of The Soul Theorem Lemma 4.1.4. Let M be a complete Riemannian manifold of nonnegative curvature and C a totally convex closed subset of M with dC / 0. Let ip(p) : C —> R be the distance funtion from p G C to dC defined by ip = d(p, dC). Then for any normal geodesic 7 : [a, b] —» C, the funcion ^(7(f)) : [a, b] —» R is concave, ie. ipinioisi + a2s2)) > o.iip(y(si)) + a2ip(-y(.s2)) where a\, a2 > 0, a\ + a2 = 1, si, s2 G [a, 6]. Furthermore, suppose ipdis)) = Z on some closed interval [a,b]. Let ra : [0,1] —> C 6e a minimal normal geodesic from 7(a) £0 9C such £/ia£ d(7(a),dC) = L Let V(s) denote the unit parallel vector field along 7|[ai&] suc/i £/ia£ V(0) := T^(0). TTiera /or any s G [a, 6], t -> exp 7 ( s ) iV(s) restricted to [0,1] is a minimal geodesic Ts from •y(s) to dC. The map <p : [a,b] x [0,/] -> C defined by •ip(s,t) = exp 7 ( s ) tV(s) spans a flat totally geodesic rectangle in C. Proof. We wish to show ip is a concave function on 7. Let s G (a, 6). Let cr5 : [0,1] —> C be a minimal normal geodesic from 7(s) to dC. Set a = Z(7'(s), a£(0)). Consider the linear function /g(s) = ip(~y(s)) — (s — s) cos a. Our goal is to show ipilis)) < l — (s — s) cos a = ipijis)) - (s - s) cos a - Ms) for all s € (s - 5, s + 5) where 5 > 0. If this holds, then for each s G (a, 6) there exists an open interval 1$ about s such that for all s G Is the function ip("r(s)) is bounded above by linear functions in the set {fs}seh- Also since fs(s) = ip(f(s)) then in fact for all s G Is •0(7(6')) = min/^s) 37 Chapter 4. Proof of The Soul Theorem Hence tp o 7 is locally the minimum of the linear functions {/s}- But lin-ear functions are concave and the minimum of concave functions is concave, therefore ip 0 7 is concave. The remainder of the proof will be divided into two cases according to the size of a. Case I: a > f _ 3 C Figure 4.2: Case I. Consider the orthogonal projection £ of 7'(s) onto the vector which is per-pendicular to CT's(0). Let be the parallel translation of £ along 0$ and note that || £ || = || "f'(s) || cos (a - f) = cos (a - | ) since 7 is a normal geodesic. Now consider the curve c8(t) = exp (Tl( t )(a-5)€(t), 0 < £ < Z Since <7$(0) e A 7 ' is a point on 7 which is contained in C and (TS(Z) € dC, we may use Lemma 4.1.3 to conclude that the tangent cone C<r-S) = { » E (T<T5(0 \ K.5(!)})l exP<rs-(0 Rf C N for some t G (0, e(am))} coincides with the open half space H = {v e CMl) \ {oas{l)}\Z(v, -a's(l)) < Since £(£) is the variation field, ^\s=sCs(l) — £(i). Furthermore, £(0 is or-thogonal to cr^(Z) because we simply parallel transported along a geodesic £ 38 Chapter 4. Proof of The Soul Theorem which was orthogonal to cr^ (O). Therefore, Z(£(l),a's(l)) = | and so ^(£(0 . -<^(0) = f- This implies that <£ H and hence £(0 g" C a _ ( / ) since these sets coincide. Then by definition of the tangent cone, Cg(l) = expCT_(;)(5 — s)£,(l) is not contained in N for sufficiently small (s — s) > 0. Therefore, the endpoint of c3(t), namely c3(l) must fall on dC or outside C. Hence, d(cs(0),dC) < L[cs}. Now we know d(cs(0),dC) < L[cs] and using Rauch's Theorem we can show L[cs] < /. We will compare our n-dimensional manifold M with sectional curvature K > 0 to Euclidean n-space. Consider the normal geodesic a3 : [0,1] —* M and a corresponding normal geodesic a% in Euclidean space. Choose £(£) to be a variation field along a3 such that (£(£), cr^ (£)) = (£(£),CT^(£)). Let f0(Nt) be the first focal value of Nt along a geodesic 77(f) = expCT_ ££(£) where denotes a hypersurface exp^ui € T^MIw ± £((), || w ||< e,e > 0 is small}. Now consider the compact set {fo(Nt)}te(o,l) and let m > 0 denote the minimum of this set. Choose s so that (s — s) \\ f (4) ||= (s - s) cos (a - •§) < /0(ATt) is satisfied. Now Corollary 2.1.10 which followed part (II) of the Rauch Theorem allows us to conclude that the length of the curve cs(t) = expa._(s — s)£(4) in M is less than the length of the curve c3(t) = expg.j(s — s)^(t) in Euclidean space. But in Euclidean space cs(t) has the same length as as(t) since we simply displaced the curve and because a3 is a normal geodesic on [0,/] we have L[as] = I. Therefore L[cs(t)} = / and so L[cs(£)] < / as desired. It is useful at this point to recall our goal. We want to show i/'(7(s)) < / — (s — s) cos a. Since V(7(s)) = d(7(s), dC) < d(7(a), cs(0)) + L[cs(t)] <d(7(s),cs(0)) + / it remains to show d(-f(s), cs(0)) < —(s - s)cosa. We may use the hinge version of Toponogov's theorem to estimate d(-f(s), cs(0)). For this purpose let r denote the segment from 7(5) to cs(0). We will use Toponogov's Theorem to compare the hinge (r,^f\^iS],a — in M with the corresponding hinge denoted (f, 7, a — | ) in Euclidean space. The hinge in the manifold has side lengths as follows: L\r] =|| (s - s)t ||= (s - 3) || £ ||= (s - s) cos(a - f) 39 Chapter 4. Proof of The Soul Theorem and Lh\[s,s]] = d(7(s)>70)) = * -The hinge in the Euclidean plane has the same side lengths; namely L[f] L[T] and L[j] = L[j]. By the law of cosines we get d^(s),cs(0)) = (.s - sY + (s - aY cos 2 ( a - - ) - 2(.s - s)(($ - s) cos (a — —))(cos (a r)) (s - 8 (s - s) cos (a--) = ( S -5) 2 ( s in 2 (a - - ) ) = (s - s)2(-cos a) 2 Using Toponogov's Theorem (B) we may conclude that d(-y(s),cs(0)) < d(>y(s),c3(0)) = -(s - s)cosa Therefore, ipi'y(s)) < I + d('y(s),c3(0)) < I — (s — S ) C O S Q which establishes the theorem for Case I. Case II: a < f Figure 4.3: Case II. Let T be the minimal geodesic from j(s) to erg. We have r perpendicular to Cg and let (Js(t0) be the point of intersection of T and a3. Since r X a3, we may apply Case I to r and 0s|[tO)(] and conclude d{1{H),dC)<L{cs)<L{a-s\[toA) = l (4.1.1) 40 Chapter 4. Proof of The Soul Theorem The first inequality holds because as previously defined, c s(0) might not fall between 7 and dC but rather somewhere on the other side of 7. The last inequality follows again from Rauch's theorem as in the proof of Case I. Now since d( 7 ( s ) ,9C) < I — t0, it remains to show that tQ > (s — S ) C O S Q . In order to do this we will consider the consequences of Toponogov's The-orem on two different hinges in the manifold. First,we can compare the hinge (7|(s,s), 0s|[o,to]!a) with the corresponding hinge (7,6>, a) in Euclidean space. The lengths of 7, and are (s — s) and ta respectively; the same as the corresponding sides in the manifold. Using the law of cosines in Euclidean space we have d2(a3(t0), 7 (s) ) = (s - s)2 + t20- 2(s - s)t0 cos a Toponogov's Theorem implies d2(as{t0)n(s)) < (s-s)2+t20-2(s-s)ta cos a. (4.1.2) Now considering the hinge (r, o"5|[o,t0], f ) in the manifold, the corresponding hinge (f, ds\\0, t0], | ) in Euclidean space, and applying Toponogov's theorem we have (s - s) < Vd2h(s),as(to)) + t2 (4.1.3) Substituting (4.1.3) into (4.1.2) we obtain d2{a-s(to),l(s)) < (d2(-/{s),a^t0)) + t2Q) + t20 - 2(s - s ) t 0 cosa which simplifies to 2(s - s)tD cos a < 2t2 and so we conclude t0 > (s — s) cos a. From (4.1.1) and the estimate of tQ above we may conclude d( 7 ( s ) ,9C) < I — t0 < I — (s — s) cos a. This establishes the theorem for Case II. Therefore, from the two cases above we have •0(7(s)) < I — (s — s )cosa for any a. Hence, ip is a concave function on 7. Now suppose ip(l(s)) = I o n some closed interval [a, b]. Let s € [a, b] and let p be a point on dC closest to 7 ( s ) . Since all points on 7|[a,h] are a distance I from dC, any point on 7I [<,,(,] other than 7(5) is a distance at least I away 41 Chapter 4. Proof of The Soul Theorem from p. Let — a3 be a minimal geodesic from p to 7 ( s ) and let a3 denote the same geodesic parametrized from 7 ( s ) to p. According to the first variation formula, — a3 meets j\[a<b] orthogonally. Hence a = Z(7'(s), 0^(0)) = f • According to the proof of case I, a = ^ implies that the curves cs(t) = expCT5( t)(s — s)£(t), 0 < t < I have length L[c s] < I. The point ca(7) lies on the geodesic r(s) = expCTs(()(s — s)£(0 and since Z( 7 ' (s) , 0^(0)) = f, then by definition of £ we have Z(£(l), o~'s(l)) = ^ . But lemma 4.0.16 implies that since g {v\/L(v, -a«(0) < f } then £(0 0 C p \ {o p} and hence cs{l) g N. Therefore, c3(l) must lie on dC or outside of C and since all points on 7|[a;,] are at a constant distance I from dC, we must have L[cs] > L Hence L\cs] = I and cs(l) G dC. In particular, since the curve cs(t) realizes the distance from c s(0) to cs(l) it is a geodesic which is contained in C by convexity. Now according to Remark 2.1.1 of the second Rauch Theorem, F(s, t) : [a, b] x [0,1}-* C defined by F(s,t) = e x p a s ( t ) ( s - s)£{t) as illustrated in Figure 4.4 is a flat totally geodesic rectangle. dC Figure 4,4: F(s,t) : [a,b] x [Q,l] -» C Let V(s) be the unit parallel vector field along 7|[<,,(,] where V(0) = cr'a(0). To prove the remainder of the theorem is suffices to show that F(s, t) and ip : [a, b] x [0, l]-*C defined by <p(s,t) = e x p 7 ( s ) tV{s) 42 Chapter 4. Proof of The Soul Theorem as shown in Figure 4.5 coincide. dC V(s) 7(a) 7(5) 7(6) Figure 4.5: ip{»,t) : [a, b] x [0,1] -> C Let V denote the induced connection on R where R is the flat totally geodesic rectangle defined by F(s,t). The previous argument which showed a = I may be applied to any cs(t) for s G [a, 6], therefore c'S(0) is orthog-onal to 7'(s) for all s G [a,b]. Since cs(t) is a geodesic of length Z which is parametrized on [0,/J, |c(.(£)| = 1 for all t. In particular |c (^0)| = 1. This implies c's(Q) is a unit parallel vector field in R for s G [a, b] or in other words Vji.(cj,(0)) = 0. But since R is totally geodesic, IIR(c's(0),c's(0)) = 0 and ds therefore V i ( 4 ( o ) ) = v | ( c ; ( o ) ) + /^(c;(o) ,4(o)) = o. Hence c's(0) is also a parallel vector field in M and by uniqueness, the geodesies which define the rectangles F(s,t) and f{s,t) must be the same. Therefore, <p(s, t) spans a flat totally geodesic rectangle in C. • We now return to the proof of Theorem 4.1.1. Proof of Theorem 4.1.1. Suppose Ca = {p G C\d(p,dC) > a} is not totally convex. Then there exists a geodesic 7 : [0,/] —» C with 7(0), 7(Z) G Ca but for some s G (0,/), 7(s) S- C a . By the definition of Ca this implies d(~f(s),dC) < a. By the definiton of the distance function ip(t) : C —» K, 43 Chapter 4. Proof of The Soul Theorem this is equivalent to the condition that ip(j(s)) < a. But since the end-points 7(0), 7(7) are contained in Ca, we have ^(7(0)) > a and >^(7(7)) > a -Therefore, there exists a minimum of 4>{^{t)) in (0,/). This is a contradic-tion since by the previous lemma ip is a concave function on 7 and concave functions do not contain a strict interior minimum. This proves part (1) of the theorem. To prove the second part of the theorem we recall that by definition of C m a x we have C m a x = p| C a = p| {PeC\d(P,dC)>a} HCV0) = {p G C\d(p,dC) = max} where max = sup{d(p, dC)\p G C). Therefore, all points in C m a x are equidistant from the boundary of C. Hence, ip is constant on any geodesic 7 in Cmax..By Lemma 4.1.4, minimal geodesies from any point on 7 to dC are perpendicular to 7. Since there is a flat totally geodesic rectangle conatined in C between C m a x and dC it follows that dim C m a x < dim C. • We are now equipped with all the tools necessary to construct the soul of M. For this purpose it will be useful to recall two main results which we will include here for clarity. (1) Any complete noncompact Riemannian manifold M of nonnegative cur-vature contains a family of compact totally convex sets Ct associated to each p G M satisfying t2 > h =*• Ch D Ctl, Ch = {q eCt2\d(q,dCt2) > t 2 - h ) Ut>0Ct = M pedC0 (2) Any closed totally convex set C contains a set N C C which has the struc-ture of a smooth imbedded totally geodesic submanifold such that C C N. To construct the soul (S), let p G M and consider the compact totally convex set CQ as defined in result (1) above. Now consider the sets 44 Chapter 4. Proof of The Soul Theorem Cl = {p € C0\d(p, 3C0) > a}, C G m a x = f l c ^ f l C° Since dC0 0 and C Q is compact, Theorem 4.1.1 implies that C " is totally convex and d i m C ™ a x < d i m C 0 . For ease of notation, set C ( l ) := C™3*. So C ( l ) is a closed totally convex set with d i m C ( l ) < d i m C 0 . If dC(l) ^ 0 then by applying Theorem 4.1.1 again, this time to C ( l ) , we obtain C(2) := C ( l ) m a x . C(2) is a closed totally convex set and dimC(2) < d i m C ( l ) . Now either dC{2) ^ 0 or dC(2) = 0. If dC{2) ^ 0 we may iterate this process letting C(i + 1) = C ( i ) m a x until we find a closed totally convex set C(k) with dC{k) — 0 or dimC(fc) = 0. This will require a finite number of steps since each time we have d i m C ( i + l ) < d imC( i ) . By construction we obtain a sequence of compact totally convex sets Ca D C ( l ) D C{2) D ••• D C(k). If dC(k) = 0, then by result (2) of the note above it must contain a set W which has the structure of a smooth imbedded totally geodesic submani-fold satisfying C(k) C N. Now since N C C(k), C(k) C N, and dC{k) = 0, we have ./V = C(k). Therefore, C(k) is a compact totally geodesic subman-ifold without boundary and dimC(k) < d i m C 0 < d i m M . On the other hand, if dimC(fc) = 0 then the set consists of a single point. Thus in either case; C(k) = S, the soul of M. 4.2 Finding a Diffeomorphism The second part of The Soul Theorem states that if 5 is the soul of M then M is diffeomorphic to the normal bundle of S. Although the exponential map is a smooth surjective map onto M , it is not necessarily injective [15]. Therefore some work is required to show the existence of a diffeomorphism between M and the normal bundle of the Soul. Before we begin the proof, we require the following definition of a critical point for the distance function. For the remainder of this paper we will denote the distance from all points q in M to a point p in M by dp(q). Definition 4.2.1. q ^ p is a critical point of dp if for any unit vector u £ TqM there exists a distance minimizing geodesic 7 from p to q which satisfies Z(u,-j'(d(p,q))) < f. Given this definition we may now prove the following proposition. Proposition 4.2.2. M \ S is ds critical point free. 45 Chapter 4. Proof of The Soul Theorem Proof. Let q 6 M \ S. Recall ct=n )^c=n(UB )^))c 7 7 t 7 In particular from Proposition 3.0.11, the Ct's expand to give all of M . This implies q £ 0Cto for some t0 > 0. Since the boundary of C j 0 is nonempty, this set may be contracted by the procedure outlined in the previous sec-tion until we arrive at the soul of the manifold. Hence q is a point in the boundary of a compact totally convex set C which contains S. We wish to show that for every minimal geodesic from S to q, there exists a unit vector in TqM satisfying Z(u, — i'{S,q)) > | . But since q G dC and S in the the interior of C , by definition 4.1.2 any minimal geodesic between them must have its initial tangent vector in the cone Cq. By Theorem 4.1.3, The cone coincides with the half space so if 7 is any minimal geodesic from q to S then all initial tangent vectors to minimal geodesies between q and S lie in H = {veTaM\Z(v,y'(0))<%} Now consider -7 '(0). Since Z{v,~/'(0)) < § then Z(v, -f{0)) > f and hence —7'(0) H. Therefore, —7'(0) makes an angle > | with every initial tangent vector to geodesies between q and S. Hence, q is not a critical point of ds. • We will also find useful the following results from the theory of ordinary differential equations. We will not provide the details of a proof here but instead the reader is referred to [12]. L e m m a 4.2.3. [12] Let M be a complete manifold and A a closed subset of M. Then for any non-critical point q of there exists a unit vector field X on some open neighborhood U of q such that < X | r , c ' ( 0 ) ) > | (4.2.4) for any r £ U and any minimal geodesic c from r to A. We call X a gradient-like vector field for the distance function. This lemma guarantees the existence of a local gradient-like vector field on M and the following corollary provides us with a global gradient-like vector field on the manifold. 46 Chapter 4. Proof of The Soul Theorem C o r o l l a r y 4.2.4. [12] Let M be a complete manifold and A a closed subset ofM. Then a) The set U of non-critical points of dA is open. b) There exists a gradient-like vector field for dA on the open set U of non-critical points. The result of the lemma which follows is particularly important because it will allow us to eventually construct a map which is necessarily injective. L e m m a 4.2.5. [12] Let M be complete, A a closed subset, U and open sub-set of M and X a gradient-like vector field for dA on U. Then a) dA is strictly increasing along any integral curve of X. b) On any compact subset C of U the increasing rate is controlled by a Lipschitz constant. We are now ready to construct a diffeomorphism between the manifold and the normal bundle of the Soul. The proof presented here is a combina-tion of those outlined in [12], [14], and [16]. Since M \ S does not contain any critical points of d$, by. Corollary 4.2.4 there exists a vector field X on M \ S such that (X\r,c'(0)) > f for any r G M \ S and any minimal geodesic c between r and S. Choose e > 0 so that the normal exponential map of 5 is a diffeomorphism from ut = {v G TpM\p G S,v ± TpS, || V || < e} onto a neighborhood Nt of S. The gradient vector field Vds(p) = ~f'(ds(p)), where 7 is the unique minimal geodesic from S to p, must satisfy the properties of X and therefore we may assume that X\NC is in fact the gradient vector field. Therefore we have a smooth vector field on M \ S which coincides with the gradient vector field on a small neighborhood of S. Consider the flow <pt generated by X. By Lemma 4.2.5, ds is strictly increasing along any integral curve of X and since the rate of increase on any compact subset of M \ S is controlled by a Lipschitz constant, the integral curves are defined for all time. Therefore $ : u(S) -* M defined by 47 Chapter 4. Proof of The Soul Theorem where v is a unit vector is a diffeomorphism. The injectivity of the map follows from the fact that the intgral curves of X do not intersect, and $ is surjective since the integral curves are defined for all time. This completes the proof of The Soul Theorem. C o r o l l a r y 4.2.6. Let M be a complete noncompact n-dimensional Rieman-nian manifold with sectional curvature K > 0. Then the soul of M is a point and M is diffeomorphic to R™ It is also interesting to note that Guijarro has since discovered that given a complete noncompact Riemannian,manifold of nonnegative curvature and soul S, it is always possible to modify the metric into one of nonnegative sectional curvature whose normal exponential map of S is a diffeomorphism [8]. 48 Chapter 5 Final Comments In this final chapter we will briefly discuss some characteristics of the soul of a manifold and then return to a more general discussion of the classification of the structure of manifolds. 5.1 Properties and Examples of a Soul It is possible for a manifold to contain numerous souls since the construction of a soul depends on choosing a point p G M. Although different choices for the initial p may lead to different souls, Y i m [18] has shown all souls within a manifold are isometric. As a trivial example, in Euclidean space every point is a soul [9]. Another example of the non-uniqueness of souls is given by the cylinder in R3. For any point p on the cylinder, the circle through p and perpendicular to the axis of the cylinder is a soul. These examples are illustrated in Figure 5.1. Figure 5.1: Examples of the non-uniqueness of souls 49 Chapter 5. Final Comments As Cheeger and Gromoll [4] have noted, the construction of a soul as outlined in this paper will not necessarily produce all submanifolds satisfy-ing the properties of a soul. For example, the paraboloid z = x2 + y2 in R3 will never arise from this construction. On the other hand, it is true that every compact submanifold S of nonnegative curvature may be realized as the soul of some M. Simply let M = S x Rk with k > 0. If M is a complete noncompact Riemannian manifold of positive sectional curvature then a soul of M is a point. This result was originally established by Gromoll and Meyer [6] but it follows directly from Lemma 4.1.4 presented here. If the soul were not a single point then it would be possible to form a geodesic in S. Since all points in S are equidistant from the boundary of a convex set C, Lemma 4.1.4 implies that there exists a flat totally geodesic rectangle in M. This is impossible since the curvature is strictly positive on all of M . The paraboloid z = x2 + y2 in R3 contains precisely one soul; the vertex. 5.2 The Soul Conjecture W i t h the establishment of the Soul Theorem, Cheeger and Gromoll [4] pro-posed that it may be possible for the soul of a manifold to be a point even when the curvature condition is relaxed. They believed that rather than strictly positive curvature everywhere on M, it is enough to have nonneg-ative curvature everywhere and positivity at a single point. This problem became known as The Soul Conjecture. The Soul Conjecture remained an open problem for more than 20 years after its initial proposal in 1972. In 1994, G. Perelman proved that the conjecture is indeed true [13]. His proof involved an extended version of Rauch's Comparison Theorem due to M . Berger and the use of Sharafudivov's existence result of a distance non-increasing retraction of the manifold onto the soul. The confirmation of the Soul Conjecture provided a significant break-through in the classification of complete noncompact manifolds of nonnega-tive curvature. Every such n—dimensional manifold M with K > 0 at some point has a soul reduced to a point and is therefore diffeomorphic to R™. This finding also provided renewed attention to this area of classification and the consequences of the Soul Theorem. 50 Chapter 5. Final Comments 5.3 Recent Research Developments The Soul Theorem tells us that every complete noncompact Riemannian manifold of nonnegative curvature is diffeomorphic to a vector bundle over a compact manifold of nonnegative curvature. Naturally, the establishment of the theorem shifted the focus of study to compact manifolds. In particular many sought to answer the question: which vector bundles over souls admit complete metrics of nonnegative curvature? In fact Cheeger and Gromoll posed the more specific question: Do all vector bundles over the standard Euclidean n-sphere admit complete metrics with K > 0? Grove and Ziller discovered that the answer is yes when n = 4 and Rigas showed this is true for n = 5, but for all other cases this question remains an open problem. Progress on the original more general question has been made by K . Tapp [17] who built upon the results of Walschap and Strake to determine a differ-ential inequality related to those manifolds which statisfy the conditions of the Soul Theorem. In particular, this inequality must be satisfied by a con-nection, a tensor, and a metric on the base space, admitted by the manifold. This provides some insight into the structure of this family of manifolds but there is still a lot of work to be done. Aside from this question, the study of compact manifolds has focused in particular on those with positive sectional curvature. The goal has been to find examples of such manifolds in order to develop the theory. This has proven to be a surprisingly difficult task! In fact, the spherical space forms with constant K = 1 and the rank one symmetric spaces (complex projective space CP™ of dimension 2n, quaternionic projective space HP™ of dimension 4n, and the Cayley plane) with \ < K < 1 are the only known simply connected manifolds of positive sectional curvature and dimension greater than 24 [7] . In addition to these examples, recent research has found certain manifolds with high degrees of symmetry that carry positive sectional curvature in dimension less than or equal to 24 (and greater than 2). In dimension 2, we have a complete classification since the Gauss-Bonnet Theorem tells us that compact manifolds of positive sectional curvature are either S2 or M P 2 . But in general, the rarity of examples continues to pose a challenge to those seeking a better understanding of the structure of these manifolds. There are however, plenty of examples available of compact man-ifolds of nonnegative curvature. Such as the higher rank symmetric spaces as well as many non-symmetric examples. It is yet to be determined if it is possible to deform the metric in any of the above examples into one of pos-itive curvature. In a related conjecture Hopf claims that although Sk x Sl, 51 Chapter 5. Final Comments k, I > 2 has a metric of nonnegative sectional curvature, it does not carry a metric of strictly positive curvature. This has been an open problem for the past 60 years and remains unsolved today. The difference in number of known examples of positively curved man-ifolds versus those of nonnegative curvature is an indication of how little is truly known about their general structure. Currently the only constraints on the topology of compact manifolds of positive curvature which do not carry over to the nonnegative case are results by Bonnet and Myers, and Synge, on fundamental groups. In odd dimensions, a compact manifold of positive curvature is orientable with a finite fundamental group. In even dimensions, if a compact manifold of positive curvature is orientable then it is necessarily simply connected. This is likely to remain an active research area for a long time to come. Around the time the Soul Theorem was established, trends in research showed increased attention to what other types of curvature; namely scalar curvature and Ricci curvature, can tell us about the topology of a mani-fold. In particular, effort has been made to generalize the classical study of the variational properties of geodesies to higher dimensions. For example, minimal hypersurfaces are being used to study manifolds of positive scalar curvature. Work in this area is continuing today. As well, the use of the Ricci flow to gain an understanding of a manifold's structure is a technique which is increasingly being recognized for its usefulness. The recent resolu-tion of the Poincare conjecture due to Perelman and work by Hamilton is the most celebrated application of this technique. Clearly the classification of the structure of manifolds of nonnegative cur-vature is far from complete. The study of the relationship between curvature and topology will likely remain an exciting area of research in Riemannian Geometry. 52 Bibl iography [1] M . Berger, A Panoramic View of Riemannian Geometry, Springer, 2003 [2] S. Brendle, R. Schoen, Manifolds with |- Pinched Curvature are Space Forms, arXiv:0705.0766v2 [Math.DG], May 2007 [3] J . Cheeger, D. Ebin, Comparison Theorems in Riemannian Geome-try, North-Holland Publishing Co., Amsterdam, 1975, North-Holland Mathematical Library, Vol.9 [4] J . Cheeger, D. Gromoll On the structure of complete manifolds of non-negative curvature, Ann. of Math. 96(1972), 413-443. [5] M.P . Do Carmo Differential Geometry of Curves and Surfaces, Prentice Hall , New Jersey, 1976 [6] D. Gromoll, W. Meyer On Complete Manifolds of Positive Curvature, Ann. of Math. 90(1969), 75-90. [7] K . Grove, B . Wilking, W . Ziller Cohomogeneity One Man-ifolds with Positive Sectional Curvature, Banff International Research Station, 04rit525 Cohomogeneity One Manifolds with Positive Sectional Curvature, Final Report, available at http://www.birs.ca/workshops/2004/04rit525/report04rit525.pdf, [8] L . Guijarro, Improving the Metric in an Open Manifold with Nonneg-ative Curvature, Proceedings of the American Mathematical Society, Vol. 126, No. 5 (1998), 1541-1545. [9] L . Guijarro, G. Walschap The Metric Projection onto the Soul, Trans-actions of the American Mathematical Society, Vol. 352, No. 1 (2000), [10] H . Karcher Riemannian Comparison Constructions, Global Differen-tial Geometry, 170-222, M A A Stud. Math., 27, Math. Assoc. America, Washington, D C , 1989 2004 55-59. 53 Bibliography [11] J . M . Lee Riemannian Manifolds. An Introduction to Curvature. Grad-uate Texts in Mathematics, 176. Springer-Velag, New York, 1997 [12] W . Meyer Toponogov's Theorem and Applications, Lec-ture Notes, College on Differential Geometry, Tri-este 1989, notes available at http://wwwmathl.uni-muenster .de/u/meyer / publications / toponogov. html [13] G. Perelman Proof of the Soul Conjecture of Cheeger and Gromoll, J . Differential Geometry 40(1994), 209-212 [14] P. Peterson Riemannian Geometry, Graduate Texts in Mathematics 171. Springer, New York, 1998 [15] W . A . Poor, Jr. Some Results on Nonnegatively Curved Manifolds, J . Differential Geometry 9 (1974), 583-600 [16] T. Sakai Riemannian Geometry, Translations of Mathematical Mono-graphs Vol. 149, 1996 [17] K . Tapp Conditions for Nonnegative Curvature on Vector Bundles and Sphere Bundles Duke Math Journal. 116, N o . l , (2003), 77-101. [18] J .W. Y i m Space of Souls in a Complete Manifold of Nonnegative Cur-vature, J . Differential Geometry 32(1992), 429-455. 54 


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