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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 THE REQUIREMENTS FOR T H E DEGREE OF 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 noncompact 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  ii  Table of Contents  iii  List of Figures  iv  Acknowledgements  v  1 Introduction  1  1.1  The Soul Theorem - Significance 1.1.1 Existence of a Soul 1.1.2 Generalization of Simple Points  2 Comparison Theorems 2.1 2.2  Rauch Comparison Theorem Toponogov's Theorem  3 3 4  6 6 18  3 Preliminaries for the Proof of The Soul Theorem  21  4 Proof of The Soul Theorem  34  4.1 4.2  Constructing the Soul Finding a Diffeomorphism  5 Final Comments 5.1 5.2 5.3  Properties and Examples of a Soul The Soul Conjecture Recent Research Developments  Bibliography  34 45  49 49 50 51  53  iii  List of Figures 2.1 2.2 2.3  Smooth variation of 7 A geodesic triangle A geodesic hinge  7 19 19  3.1 3.2 3.3 3.4  Geodesic Cone Construction of H Proof that d(q, B ) < t - h Hypersurface and corresponding geodesic cone  23 26 30 32  4.1 4.2 4.3 4.4 4.5  Parallel Translation Case 1 Case II F(s,t) : [a,b] x [0,/] C <p{s,t) : [a,b] x [0,1] -» C  36 38 40 42  5.1  Examples of the non-uniqueness of souls  49  1  li2  2  43  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 Riemannian Geometry. Early investigations focused on the simplest case, namely those manifolds which possessed constant sectional curvature. A tremendous 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 ) when K = 0, the n-Sphere (S ) of radius when K > 0 and n- dimensional hyperbolic space (H ) 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. n  n  n  Following the classification of manifolds admitting a metric of constant sectional curvature, the next step was to attempt to classify arbitrary Riemannian maniflods. Since much was known about the structure of manifolds of constant sectional curvature, a natural approach was to seek a classification of these manifolds through comparison with model spaces. The hope was that if information such as bounds on K were known it would be possible 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 Riemannian 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 extremely 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 manifolds of nonnegative curvature must contain points at which the sectional curvature gets arbitrarily close to zero. B y exploiting the Gauss-Bonnet formula specified for surfaces in R , Cohn-Vossen determined the structure of all such manifolds in dimension 2. In particular, any 2-dimensional noncompact complete manifold with K > 0 is either diffeomorphic to M. 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 allowed them to determine the structure of Riemannian manifolds when the curvature is strictly positive. Their remarkable result is that any such ndimensional noncompact manifold is diffeomorphic to 1™. A crucial step in their novel approach which allowed for classification involved associating 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 entire 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 manifold 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. 2  2  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 Theorem 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 diffeomorphism 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 particular 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 particular the usefulness of the study of totally convex sets in understanding the structure of manifolds. Following the proof of the theorem we will return 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 classification of noncompact manifolds of nonnegative curvature. The statement of the theorem itself is quite surprising and in addition it provides a generalization 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 demonstrated 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 on M and the induced connection V on S and suppose X and Y are vectorfieldstangent to S and extended arbitrarily to M. We may write M  5  v^y = (v^r) + (v^y) T  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 vectorfieldstangent to S we have V%Y  = (X7 Y) S  X  +  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]. Gromoll 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 computation 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 describes 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 Jacobifieldsas 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) = 7 (i) and F(0,t) = 7 (t) = j(t). F is a variation through.geodesies if each 7 (£) is a geodesic. Let T = 7% 7«(0 f let V = denote the variation field. A variation of 7 is illustrated in Figure 2.1. s  0  S  =  a n (  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 vectorfieldalong 7. // V is the variationfieldof 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 vectorfieldalong 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 T M, 7 satisfying initial conditions p  J(a) = X  there exists a unique JacobifieldJ along  and  V J(a) = Y. t  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 Jacobifieldalong 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 = exp v. 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 : T M —» M. p  p  The second statement is the more intuitive one; conjugate points q = exp t) are the points where exp fails to be a local diffeomorphism at v. For example, let p be any point on the unit sphere in R . 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 exp 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 do not contain any conjugate points but once you travel more than halfway around a circular geodesic it is no longer minimizing [11]. p  p  3  p  3  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 M and W £ T M. A unique 7 ( a )  y{b)  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 € T M is said to be normal to S if for every v G T S, (n, v) = 0. The set of all n 6 T M which satisfy this property is called the normal space to S at p and is denoted by N S or TpS- . The normal bundle NS is a vector bundle over S defined by NS = Upgs N S. This is similar to the tangent bundle TM to M at p which is defined as TM - \J T M. 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 |jvsP  p  p  1  P  P  peM  p  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  II i ll=i. Definition 2.1.6. For any p £ M, the geodesic submanifold defined by v € T M is exp (U) where U is a neighborhood of the origin in {w £ T M\w ± v} for which expp \ u is an embedding. p  p  p  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 M be Riemannian manifolds. Supa  pose dimM < dimM . Let 7 : [0, /] —> M and j : [0,1] —> M be normal geodesies. Assume K(o ) > K(o) for all plane sections o , a along 7 and 7 containing 7^ and 7' respectively. Q  0  a  0  Q  D  (I)Assume further that for any t £ [0,1], "fo(t) is not conjugate to 7 (0) along j . Let V and V be Jacobi fields along 7 and j satisfying the following conditions (a) V(0) and V (0) are tangent to 7 and j o  0  Q  a  o  0  (b) || V(0) || = || V (0) II and \\ V'(0) || = || V '(0) || (c) (T(0), V'(0)> = (T (0), V'(0)> where T = 7', T = i o  o  o  o  0  Q  Then for all t e [0,1],  9  Chapter 2. Comparison Theorems  II V(t) \\>\\v {t) ||. 0  (II) Assume that for any t G [0, Z], "f(t) is not a focal point of the geodesic submanifold N defined by T . Let V and V be Jacobifieldsalong 7 and "f satisfying the conditions (a) V(0) and V (0) are tangent to 7 and-f Q  D  a  o  0  0  (b) || V(0) || = || V (0) I and \\ V'(0) \\ = \\ K '(0) || (c) (T(0),V(0)) = (r (0),V- (0)> o  o  o  o  Then for all t G [0,/], I V(t) \\>\\V (t) ||. 0  The following definition and lemma play a crucial role in the proof of the Rauch Theorem. Let 7 : [a, b] —> M be a geodesic. D e f i n i t i o n 2.1.8. The index form I(V,W) given by W  )  W  = Sa^rV, V W) T  +  (R(W,T)V,T)  is a symmetric bilinear form on all smooth vectorfieldsV 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 vectorfieldswhich 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 vectorfieldon 7 and V is the unique Jacobifieldsatisfying V\ = W\ = 0 and V\ = W\ . Then p  q  p  q  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 T M. By the existence and uniqueness theorem for Jacobi fields we may extend each Vi to a unique Jacobi field along 7 satisfying Vi\ = 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 q  p  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\ = 0 and V\ = W\ we have V = ^ fi{l)Vi. Using these definitions for W and V we will now determine expressions for p  q  q  /(V, V) and I{W,W).  I(W)=  f\v',V') + (R(T,V)T,V)  Jo  = / (V,vyJo  (R(T,V)T,V)  (v",v) +  = / (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/ (i)) t  i  i  We wish to show I(V, V) < I(W, W) so we now turn our attention to evaluating I(W,W). By definiton I(W, W) = ^{{W, W) + (R(T, W)T, W)) and using the definition of W we expand the first term in the integral as follows f\w\w')=  A(£/^)',(]T/^)'>  Jo  Jo  i  '(EiflVi  + fiVJ),  i  ' 0  + jwi)) i  i  i  i  i  ii  Chapter 2. Comparison Theorems  For simplicity we will evaluate the terms in the above integral separately.  =  T,[ fiMM,v )'-M',v )) 1  j  id  j  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) = y> ) - E [\f&M, 1{  v}>+v ))  v  -E  f i,j  3  fifiWTMTtVj) 0  = /(V, V ) - E  fififiW,  V ) + hf'jiVl, Vj)) 3  W . E / i W E ^ - )  = /(v, v) - E fu'iliW, v ) + f.ftiv;, 3  -  f  Vj))  (R(T,W)T,W)  Jo  =i(v,v)-  A E ^ E ^ >  - JoA Ei - ^ Ei - ^ -  Jo  l\R{T,W)T,W)  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• ^ ' - Ej ^ ) - JoA E ^ ' ' E ^ >  Jo  l  f (R(T, W)T, w) + f Jo Jo  3  (E m, E fiV() i  +A E / ^ E ^ > + AE/^'EZ^) 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  i  0  Wi)  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  fi *> E V  fi i V  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)  - (v vj)y u  = (v(', Vj) + (v;, vj) = (VlWj)  - (v(, vj) - (v , vj') t  - (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  For now we will assume that V, and V , are perpendicular to T, and T or that || V(0) ||= (r(0),V'(0)) = 0 and || V (0) ||=<r (0),V- '(0))=0. Now consider the ratio |y where t is the same parameter used to define the geodesies 7 and 7 . As in the statement of the theorem, 7 contains no points conjugate to 7 (0) along 7 so the ratio is defined for all t except t = 0. Using L'Hopital's rule we may evaluate the limit Proof of the Rauch Theorem (I).  0  0  o  o  o  G  G  o  l i m  iL|.  t_o I V ||  Q  l i m  i^L  = lim  t-o (V , V )  2  0  0  |ir^  t-,o 2(V ', V„)  0  0  = lim < "> ) + <y'' ') V  V  (212)  V  These terms vanish at t=0 and so from the above calculation the limit simplifies to || v ||  {V{o),v'{o))  2  ^°nm  in i Q\  I no) I _ 1  =  (no),no)>  2  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 \\>\\ V || will follow immediately. Since 0  v II\  _ (V ,Vo)i(V,V)-(V,V)i(V ,Vo)  2  0  0  dt\\\V fJ  (i/ ,V )  0  0  >  0  2  0  if and only if 2{V ,V )(V',V) - 2(V,V)(V^,V ) > 0 0  0  0  if and only if (V ,V )(V\V)>(V,V)(V:,V ) 0  0  0  if and only if (V',V) (V,V)  we will show  >  -  (VLV ) 0  (V ,V ) 0  0  is increasing by showing ^'yj >  ^  o ra u  ^ > 014  Chapter 2. Comparison Theorems  Let s e [0,1) and consider the vectorfieldswhich are constant multiples of V and V defined by Q  By the construction of W (t) it is clear that s  (V, V) _ (W , W' ) s  (V,V)  S  (VI, V ) _ (W' , W )  s,riQ  s  (W ,W )  0  0s  (V ,V )  S  0  (2.1.4)  0s  (W ,W )  0  0a  0s  1, when evaluated at t = s, the relation on the  Since || W,(s) ||= left hand side becomes (V, V'\  II  (V,V)  (2.1.5)  W {s) || a  (2.1.6)  [ (w ,w' S  s  Jo  (2.1.7)  s  f(w>,w' ) + (w';,w ) f s  Jo  -  (2.1.9)  (R(W ,T)T,W ) S  S  Jo f (W' ,W' ) - K(o) || W Jo S  (2.1.8)  s  S  (2.1.10)  s  where er is the plane section spanned by T and since W is a constant multiple of a Jacobi field.  >  an<  ^ (2-1-9) follows  s  Let P- denote the parallel translation along 7 in the direction from q to p and let P be the parallel translation along j . Define I : T ^M —> T ( )(M ) so that I is an injection which preserves the inner product and It °: T M -» T (M ) by I (X) = P o / o P_ (X). In addition, we will assume h(T) = T and / (W ) = W . Finally, define a field W by 1  lo  7o  0  0  y  o  7 ( t )  lo(t)  0  t  0  S  lo  5  7  0s  0s  W^ (t) = h(W (t)) a  s  Since 7 preserves the inner product (W (t),W (t)) s  s  = (W,,.(t),W.(0)) o  and (Wfr),W (t)) a  15  Chapter 2. Comparison Theorems Given these new structures and since we know K(o ) > K{o) we may proceed with the calculation given in (2.1.10) 0  (VX)  (W' , W' ) - K{o) || W f s  (W)  s  (2.1.11)  s  Jo (W> ,W> ) - K(o) || W a  i:  s  > / (W ,W> )-K(CT )\\W Jo 03  S  0  = [\w^ ,W^J s  Jo  (2.1.12)  Qs  ||  (2.1.13)  2  03  (R (W7,,T )T ,W; ) 0  0  0  (2.1.14)  S  (2.1.15) (2.1.16)  > I{W ,Wo ) by Lemma 2.1.9 03  s  = /V;,<>-  (2.1.17)  (R(W ,T)T,W ) 0s  0s  so we have (V,V) (V,V)  > I*'«,,<,)Jo  {Ro(W ,T)T,W ) 0s  0s  = /Vo,.W^.>-«.^o.> Jo  =  f (w „w ,y s  0  0  Jo =  « Xo )\s s  _  a  (VoX) using (2.1.4) (Vo, V ) 0  Since this holds for any 5 € [0,1], we have desired.  > (vTy') for all t as  We have shown the theorem holds for the specific case when V and V are perpendicular to T and T or when both (T(0), V"(0)) and (T (0), V '(0)) equal zero. Now we must consider the general case.  Q  Q  o  o  Decompose V and V as follows: Q  V = V+(T,V)T  and V = V + (T , V )T 0  0  0  0  0  16  Chapter 2. Comparison Theorems where V and V satisfy the conditions of the first case. Therefore, || V(t) \\>\\ V \t) || holds for all t. Now since 0  0  T(T, V) = ( V T , V') + (T, V V) T  T  = (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 {T , V ) = (T (0), V '(0))t + (T (0), V (0)> 0  0  o  o  o  (2.1.19)  o  B y assumption (T(0), V"(0)> = (T (0), V '(0)> and (T(0), V(0)) = (T (0), V (0)> therefore we have (T,V) = (T ,V ) from (2.1.18) and (2.1.19). Hence from the decomposition we may now conclude o  0  o  o  o  0  || V(t) \\>\\V (t) [[ for all £ e [0,Z]. 0  • 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 vectorfieldsalong 7 and 7 which satisfy (E(t),f(t)} = (E(t), ^'(t)). Let Nt denote the hypersurface 17  Chapter 2. Comparison Theorems  exp ( ){u; € T ( )M|u; _L E(t), \\ w \\< e,e > 0 is small} 7 (  7  t  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 K > K^, for all plane sections a, a in M and M respectively. Then a  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) = exp ( ) f(t)E(t) determine a flat totally geodesic rectangle in 7 t  M.  2.2  Toponogov's Theorem  Toponogov's Theorem is a global generalization of the first Rauch Comparison 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 Theorem are as follows: Definition 2.2.1. A geodesic triangle (71,72)73) in a, Riemannian manifold M is afigureconsisting of three distinct points p\, p , 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. 2  +  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 (i ),y (0)). ,  1  1  2  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 determined 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 minimal, 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(/ )) < d(7i(0),72(^2)) 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 manifold 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 provide 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 P r o o f of The Soul Theorem Totally convex sets play a fundamental role in the proof of the Soul Theorem 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 manifold 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 ( j ( p ) is strongly convex, then C is said to be locally convex or simply convex. t  p  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 B ^(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]. r  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 {N } A be the collection of all suchfc-dimensionalsubmanifolds and take N = \J N . a  aG  a  a  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 — > R . Now since p £ N, we must have p £ N for some a. We know N is a submanifold of M , so there exists a coordinate neighbourhood V of p in N and a diffeomorphism ip : V — > R . Without loss of generality we may choose V C B^. It suffices to show that the fc  a  a  fc  a  2  chart (V,ip) on N 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 a  N (V) = {v€T N± s  q  :\\v\\<6,qe V}  where 0 < S < is chosen so that the normal exponential map exp- : N$(V) —> M is a diffeomorphism onto its image. 1  By the inverse mapping theorem, since Ng(V) is an open set in the normal bundle of N , and exp- is continuous, the image T$(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 ^T (V)DN. 1  ±  a  S  Since V / T (V) n N, there exists a point q € N satisfying q € T (V) n N but g ^ V. By definition, since q € N we have g £ Np for some /3. Let r £ AT be the point closest to q. Since N and are submanifolds of M , the minimal geodesic 7 joining g and r is orthogonal to V. Then since N is a smooth submanifold and exp,j is a diffeomorphism from a neighborhood of the origin in T M to a neighborhood of r in M , there is an open neighborhood V C V of r in N , such that the minimal geodesies from q to the points in V' meet V transversally. 5  S  Q  a  a  q  a  22  Chapter 3. Preliminaries for the Proof of The Soul Theorem N  0  Figure 3.1: Geodesic Cone  We claim that the cone defined by A = {exp tv\v e T M, \\v\\ < e(q),exp v e V',0 < t < 1} q  q  q  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  T M\exp v£ q  r  V, \\v\\ < e{q),0 < t < 1}  V is an open subset of a fc-dimensional manifold, so e x p ~ ( V ) is a kdimensional submanifold of the tangent space T M, and the rays from the origin in T M meet e x p ^ ( V ) transversally. Therefore the cone H is a (k + l)-dimensional submanifold of T M, and so A = exp H 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. 1  q  1  /  q  q  q  Now we must show that N is totally geodesic. Let 7 be a geodesic beginning 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, ] is a minimal geodesic from p to e  23  Chapter 3. Preliminaries for the Proof of The Soul Theorem  = -y(e). Since M is complete there exists a minimal geodesic 7 in M between 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 B ( ), 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. • q  e  p  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 nonnegative 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, exp is defined for all v £ T M, ie. if any geodesic p  p  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 T M there exists an accumulation point v for the sequence of vectors (7,'(0)}. The existence of a ray 7 starting at p follows by defining 7 : [0, 00) —» M to be the geodesic with 7(0) = p and 7' = v. p  Since the requirement of completeness necessitates the existence of rays, the topology of a complete manifold of nonnegative curvature is greatly restricted 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 construction 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 nonnegative 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? = (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 H := M\B 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 £ B (j(t)). 7  7  7  t  We will now show, using a proof by contradiction, that for any ray 7, 7/ is a totally convex set associated with p. To begin, assume i / is not totally convex. This implies that there exists a geodesic 7 : [0,1] —> M with 7 (0), and 7o(l) G H-y, but f (s) 0 H-y for some s G (0,1). For ease of notation let q = 7o(s). Since q g" H , then in particular q G Bt("i(t)) for some t > 0. In fact, there exists a t > 0 such that q G Bt{~i(t)) for all t > t . This is true since by the construction of the metric balls along 7, i?t (7(^2)) D Bt (j(ti)) for all t\ < £27  7  0  o  0  y  0  0  2  1  Now let d(q,j(t )) — t — e for some e > 0. Then 0  D  d(q^(t))  < d(g,7(*o)) + d(7(to).7(*)) = =  t - e+t - t 0  0  t-(.  25  Chapter 3. Preliminaries for the Proof of The Soul Theorem  Figure 3.2: Construction of Hj. for all t > t since 7 is a ray. Q  For each t, let 7o(st) be a point on j which is closest to ~f(t). Consider the minimal geodesies Q  7o 7o|[o,st] =  •y\ from jo(s ) to >y(t) t  7^ from 7(i) to 7 (0) o  Since q is a point on 7 and 7 (.s't) is the closest point on 7 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 E then in particular 7„(0) 0 B {"i{t)) and so d(7 (0), 7(f)) > Therefore £[73] > d(7o(0), 7(0) ^ By definition 7J and 73 are minimal geodesies and since £[7*] < £[7o] isfinitewe have + L\y\] > Z/[7] f° sufficiently large Therefore (7^, 71,72)formsa 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 a = § since M is complete and j (st) is the closest point on f to the point 7(4) on M . G  1  0  0  o  t  0  2  0  0  26  r  Chapter 3. Preliminaries for the Proof of The Soul Theorem  Therefore, Toponogov's theorem implies that in particular a < f • If we consider the geodesic triangle in the euclidean plane, we may use the law of cosines as follows: 2  Lm 2  = Lm  +  2  L [ ] - L [^]L [y{} cosa\ 2  2  2  7l  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 t < L tf ] < L [ ] + (t - e) 2  2  2  2  2  lo  or simply t <L h } 2  +  2  0  (t-e)  2  which expands and simplifies to t < L [ ] + t - 2et + e 2  2  2  2  lo  if and only if 2te < L Yi ] + e 2  2  0  or ^L [ ]  +e  2  t  2  lo  e  But this is impossible for sufficiently large t since L[y } is fixed. Therefore we have a contradiction, and hence H is totally convex. • 0  y  Now that we have established the existence of a totally convex set associated 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 additional 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) t > t\ implies Ct D Ct and 2  2  Ct, =  x  {qeCtM<l,dC )>t -h} t2  2  27  Chapter 3. Preliminaries for the Proof of The Soul Theorem in particular, dC  tx  (8)\J > Ct t  (3)  = {q € C \d(q,dC ) t2  =  t2  t -t }. 2  l  = M,  0  edC .  P  0  Proof. Let p € M and let 7 : [0, 00) —> M be a ray starting at p. Let 7t : [0, 00) —> M satisfying t(s) = ( 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 ] H^ where the intersection is taken over all rays 7 starting at p, satisfies the requirements of the proposition. 7  7  7  t  B y the previous theorem each Hj is totally convex and closed. Therefore, 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, C is noncompact then there exists a sequence of points {pi}^i in C satisfying l i m j _ d(p, pi) = 00 Let 7, : [0,/?j] —» M be minimal normal geodesies from p to pi. B y definition, p € H for every ray 7, so p £ Ct- Therefore the total convexity of C implies that every 7; is contained in C . B y compactness of the unit sphere in T M, there exists an accumulation vector v £ TpM of the set {7^(0)} for which a subsequence of {7;} converges to a ray „ : [0, 00) —• C . Thus we have found a ray emanating from p which is entirely contained in Ct- This is impossible by the definition of C since in particular 7,, <£ . Therefore, Ct is compact. t  t  0 0  lt  t  t  p  7  t  7  v  t  Now assume t >t\. We have 2  B ( (s)) s  lt2  + t2)) = B +t -t,(7t (s + i 2 - t i ) )  = BsMs + ti)) C B - (l(s s+t2  h  8  a  l  in particular  B ht (s)) C B s  2  s+t2  _ (7 (s + t - h)) tl  tl  2  implies that  \jB ( (s))c\jB ( As)) s  lt2  s  s  lt  s  or by the notation introduced in the previous proof B~?t C B 2  yii  28  Chapter 3. Preliminaries for the Proof of The Soul Theorem  for any ray. Taking complements of these sets gives ii  = ( i,. )  H  B  2  3 (5 ) = H c  c  7(l  2  lti  and thus Ct D C 2  tl  To prove the remainder of part (1) we require C  ={q€C \d(q,dCti)>t -t }  tl  t2  2  1  We claim that it is enough to show B =  {q\d{q,B )  1H  <t -h}  ll2  2  for any 7 emanating from p. If this were true then we would have {JB  ={q\d(q,\jB )<t -h}  1H  li2  7  2  7  where 7 ranges over all rays emanating from p. Hence C  tl  = p| H  lti  1  = p|(S 1  ) = (|J B c  7li  ) = {q\d(q, [j B  )>t -t }  c  yti  lt2  2  1  1  1  (3.0.1) Now if we substitute d(q,\jB^)  =  d(q,d^[JB^)  = d(g,5^P|<)) = d(q,8C ) t2  into the final expression in (3.0.1) we obtain as desired.  = {q\d(q,DC ) > t — £1}  Now it only remains to prove the claim that B  t2  lti  = {q\d(q, B ) ll2  2  < t - ti}. 2  29  Chapter 3. Preliminaries for the Proof of The Soul Theorem  First, let q be such that d(q,B ) < t — t\. Then there must be a point q' € B such that d(q.q') < t — <i- Therefore when s > 0 is sufficiently large, q' G B (j (s))- Hence q G B t _ (7t (s)) by the triangle inequality. But we previously showed that B C B , therefore q G B . Thus yi  yt2  2  2  s  s+  t2  2  tl  7 ( 2  {q\d{q,B ) < t - t i } C yi2  then for some s > 0,  _ ( ( . s + i ~h)) = B tl  7 (  1H  lti  s+t2  yi  B .  2  On the other hand, if q G B  q GB  2  7tl  2  s+t2  _ ( (.s)) D B ( 7 ( « ) ) tl  7t2  s  ta  Therefore, d(g,J3 ) <d(<7,B (7t (s))) < * 2 - t i 7t2  a  as shown in Figure 3.3. Thus B  lti  2  C {q\d(q, B , ) < t — t\}. 7  2  2  B + t - t ( 7 t ( s + *2 -«i)) a  a  1  1  Figure 3.3: Proof that d(<j, B ) < t - ti 7(2  2  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, B ( (s))) s  lt  + d(p, q) > d(p, B { {s))) = t. s  lt  30  Chapter 3. Preliminaries for the Proof of The Soul Theorem  But d(p, q) < t implies that d{q, B (-y (s))) > 0. Therefore g G (£? (7t(s))) = H . Since this holds for all rays, q € Ct- Therefore, \J Ct = M. c  s  s  t  yt  t  Part (3) of the proposition follows immediately from the definition of C.  •  0  The proof of the above proposition follows closely the original proof presented by Cheeger and Gromoll in [4]. An alternative approach is presented separately by P. Peterson [14] and T. Sakai [16]. These proofs make explicit use of a map called the Busemann function. For any ray 7 in M , the Busemann function 6 is defined by 7  B^iq)  = lim _ t  t+0O  (i - d(q,-y(t)))  The Busemann function is defined on all of M since there exists a ray associated 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 ( ( ~ > *])} P ° 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. 1  0 0  t  o  r  v e  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 = \J N where {N } A 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. a  a  a  ae  L e m m a 3.0.12. Let C be convex and connected, and let  p'6Bi  peCnN,  £(p)  (p)nC,  qe  Bit(p)(p)nN  Let 7 be the normal geodesic in M such that 7|[o, ] * the minimal segment from q to p' where e. = diq,p'). Then 7|[o, ) C Af and hence p' £ N. If furthermore p' £ N, then f{s) £ C for all e < s < e + |e(p). s  e  e  Proof.  Let 0  <  e < e + \e(p)  elements of B^ip)  and p = 7(8) be a point in C. Since q, p' are  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 (p) ) as well. Hence p G C f | B ( ) ( p ) . e  (p  e  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 ( )(p) f]C, the strong convexity of the set implies there exists a unique minimal geodesic contained in B i ^{p)C\C between p and points in W. Hence the geodesies forming the cone must be contained in C. B y the same argument presented in Theorem 3.0.7, the geodesic cone is a smooth k- dimensional submanifold of M . But since N = ( J N where each N is a k—dimensional submanifold of M which is contained in C; this cone must be one of the AT 's. Therefore, V C N. Then in particular if we let p = p', we have |[o, ) C N since this geodesic segment is necessarily part of the cone. Hence p' G N. e  p  c p  a  a  a  Q  7  c  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 connected subspaces. Let the partition be denoted by {N }i^j where I is an index set. The strategy of the proof is to show that for any connected component N in the partition, C C N . We already have by construction that N 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. l  l  l  l  %  It remains to show C C JV . The result follows from a proof by contradiction. Suppose C <£ N\ Since C <£ N\ there exists a point p' e (C\ N ). Since C is connected we may choose p 6 N n C such that p' 6 (p) CI (C \ N ) 4 and q € B^(p) fl N . But these points satisfy the assumptions of Lemma 4 3.0.12, therefore p' € N . This is a contradiction, hence C C N • l  i  l  l  z  l  l  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 noncompact 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 convexity. 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 afinitenumber 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 C , and C as a  C = {p& C\d{p, dC) > a}, a  C  max  max  = p| C  a  Then (1) for any a, C is totally convex, a  (2) dim C  m a x  < dim C  The proof of this theorem requires that wefirstintroduce 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. D e f i n i t i o n 4.1.2. The tangent cone C at p g C is the set p  34  Chapter 4. Proof of The Soul Theorem {v £ T M\exp(j^jj) £ N for some positive t < e(p)} \J{o } P  p  where o denotes the origin in the tangent space T M. p  p  Let C denote the subspace of T M generated by C . p  P  p  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 C \ {o } is the open half-space p  H = {veC \{o }\Z(v, p  p  - '(0)<f}.  p  7  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 willfirstshow that H C C \{o }. Let v £ H and choose s £ (0,/) so that d(p,7(s)) < Since 7 is a minimal geodesic, the distance between 7(s) and dC is / - s. Therefore B;_ (7(s)) ndC = {p}. Since 7 is orthogonal to the boundary of C and v € H means Z(v, — 7'(/)) < §, we have exp contained in the interior of C for some t £ (0, e(p)). Hence, p  p  s  p  HcC \{o }. p  p  Now to show C \{o } C H we will first use a proof by contradiction to show that C \ {o } c H. Suppose v £ C \ {o } 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 element of H is an element of C \ {o }, so in particular — v £ C \ {o }. By definition, this implies exp 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_ for which "f- (e) is contained in ./V. Equivalently, (—e) is an element of N. Now consider 7 |[_ 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, j is not contained in A^ beyond p. But this is a contradiction since v £ C implies 7„(t) is contained in N for some £ £ (0, e(p)). Hence C \ {o } C H. p  p  p  p  p  p  p  p  p  p  p  w  v  7v  v  C)  v  p  p  p  To complete the proof it suffices to show C \ {o } is an open set in C . Without loss of generality, assume q £ (p) n N. Recall 7 is a minimal p  p  p  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 (- )'(e) g" C but -(- )'(e) = T'(0) G C . Hence 7  p  7  p  Let v £ C \ {o } and consider the curve c(s) = exp st>. By definition and Lemma 3.0.12, c(s) is contained in N for all sufficiently small s > 0. Let cr denote the minimal geodesic joining the point c(s) to q. Let P ^ denote the parallel translation along <J from c(s) to Then since N is totally geodesic, P {c'{s)) is contained in T N. Since v = lim_*o c'(s) it follows that p  p  p  s  S  aa  s  Q  P  C T  » = P » = lim ^o Pa (c'(s)) s  s  is contained in T N. Since this holds for any v £ C \{o }, we have P ( C ) C which implies C C P- (T N) or in general C C P_ (T,iV). p  Q  TN  P  Q  Y  p  7  P  7  P  Q  Figure 4.1: Parallel Translation To see that C \ {o } is open in P_ (T A^), consider an open neighborhood A^ ( ) of c(.s) contained in ./V and minimal geodesies from p to every r £ N / y 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 P (i») in T N. Therefore the set of initial tangent vectors forms an open neighborhood of v in P_ (T A ). Each of these initial tangent vectors is, by definition, contained in C \{o }. Thus, using this construction, for every v £ C \ {o } there exists an open neighborhood of v in P_ (T A ) that is contained in C \{o }. Hence C \{o } is an open subset in C . • p  p  7  g  c 5  c  S  7  q  r  7  9  p  p  p  p  r  7  ?  p  p  p  p  p  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 + a s )) > o.iip(y(si)) + a ip(-y(.s )) 2  where a\, a  2  2  2  2  > 0, a\ + a = 1, si, s G [a, 6]. 2  2  Furthermore, suppose ipdis)) = Z on some closed interval [a,b]. Let r : [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 vectorfieldalong |[ &] a  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 T from •y(s) to dC. The map s  <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 cr : [0,1] —> C be a minimal normal geodesic from 7(s) to dC. Set a = Z( '(s), a£(0)). 5  7  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 linear functions are concave and the minimum of concave functions is concave, therefore ip 7 is concave. 0  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 perpendicular to CT' (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 s  c (t) = exp 8  (Tl(t)  (a-5)€(t),  0< £< Z  Since <7$(0) e A ' is a point on 7 which is contained in C and may use Lemma 4.1.3 to conclude that the tangent cone 7  (TS(Z) €  dC,  we  \ K. (!)})l P<r-(0 Rf C N for some t G (0, e(a ))} coincides with the open half space <r-S)  C  =  { »  E  ( <T (0  ex  T  5  5  H = {v e C  Ml)  s  m  \ {o }\Z(v, -a' (l)) < as{l)  s  Since £(£) is the variation field, ^\ =sC (l) — £(i). Furthermore, £(0 is orthogonal to cr^(Z) because we simply parallel transported along a geodesic £ s  s  38  Chapter 4. Proof of The Soul Theorem  which was orthogonal to cr^(O). Therefore, Z(£(l),a' (l)) = | and so ^(£(0.-<^(0) = f- This implies that <£ H and hence £(0 g" C _ since these sets coincide. Then by definition of the tangent cone, Cg(l) = exp _(;)(5 — s)£,(l) is not contained in N for sufficiently small (s — s) > 0. Therefore, the endpoint of c (t), namely c (l) must fall on dC or outside C. Hence, d(c (0),dC) < L[c }. s  a  (/)  CT  3  s  3  s  Now we know d(c (0),dC) < L[c ] and using Rauch's Theorem we can show L[c ] < /. We will compare our n-dimensional manifold M with sectional curvature K > 0 to Euclidean n-space. Consider the normal geodesic a : [0,1] —* M and a corresponding normal geodesic a% in Euclidean space. Choose £(£) to be a variation field along a such that (£(£), cr^(£)) = (£(£),CT^(£)). Let f (Nt) be the first focal value of Nt along a geodesic 77(f) = exp _ ££(£) 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) cos (a - •§) < / (AT ) 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 c (t) = exp ._(s — s)£(4) in M is less than the length of the curve c (t) = exp . (s — s)^(t) in Euclidean space. But in Euclidean space c (t) has the same length as a (t) since we simply displaced the curve and because a is a normal geodesic on [0,/] we have L[as] = I. Therefore L[c (t)} = / and so L[c (£)] < / as desired. s  s  s  3  3  0  CT  s  0  s  t  a  g j  3  s  s  3  s  s  It is useful at this point to recall our goal. We want to show i/'(7( )) < / — (s — s) cos a. Since s  V(7(s)) = d( (s), dC) < d( (a), c (0)) + L[c (t)] <d(7(s),c (0)) + / 7  7  s  s  s  it remains to show d(-f(s), c (0)) < —(s - s)cosa. s  We may use the hinge version of Toponogov's theorem to estimate d(-f(s), c (0)). s  For this purpose let r denote the segment from 7 ( 5 ) to c (0). We will use Toponogov's Theorem to compare the hinge (r,^f\^ ],a — in M with the corresponding hinge denoted (f, 7, a — | ) in Euclidean space. The hinge in the manifold has side lengths as follows: s  iS  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),c (0)) = (.s - sY + (s - aY cos ( a - - ) 2  s  - 2(.s - s)(($ - s) cos (a — —))(cos (ar)) (s - s) cos  (s - 8  (a--)  = ( -5) (sin (a--)) 2  2  S  = (s - s) (-cos a) 2  2  Using Toponogov's Theorem (B) we may conclude that d(-y(s),c (0)) < d(>y(s),c (0)) = -(s - s)cosa s  3  Therefore, ipi'y(s)) < I + d('y(s),c (0)) < I — (s — the theorem for Case I. 3  S)COSQ  which establishes  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(t ) be the point of intersection of T and a . Since r X a , we may apply Case I to r and 0s|[t (] and conclude 0  3  3  O)  d{ {H),dC)<L{c )<L{a- \ ) 1  s  s  [toA  = l  (4.1.1) 40  Chapter 4. Proof of The Soul Theorem The first inequality holds because as previously defined, c (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. s  Now since d ( ( s ) , 9 C ) < I — t , it remains to show that t > (s — S ) C O S Q . In order to do this we will consider the consequences of Toponogov's Theorem on two different hinges in the manifold. First,we can compare the hinge (7|(s, ), 0s|[o,t ]! ) with the corresponding hinge (7,6>, a) in Euclidean space. The lengths of 7, and are (s — s) and t respectively; the same as the corresponding sides in the manifold. Using the law of cosines in Euclidean space we have 7  0  Q  a  s  o  a  d (a (t ), ( s ) ) = (s - s) + t 2  2  3  0  2  7  0  2(s - s)t cos a 0  Toponogov's Theorem implies  d (as{t )n(s)) < (s-s) +t -2(s-s)t 2  2  0  cos a.  2  0  a  (4.1.2)  Now considering the hinge (r, o"5|[o,t ], f ) in the manifold, the corresponding hinge (f, ds\\0, t ], | ) in Euclidean space, and applying Toponogov's theorem we have 0  0  (s - s) < Vd h(s),a (to)) + t 2  (4.1.3)  2  s  Substituting (4.1.3) into (4.1.2) we obtain  d {a- (to),l(s)) < (d (-/{s),a^t )) + t ) + t - 2(s - s ) t c o s a 2  2  2  s  0  2  Q  0  0  which simplifies to 2(s - s)t cos a < 2t  2  D  and so we conclude  t > (s — s) cos a. 0  From (4.1.1) and the estimate of t above we may conclude d ( ( s ) , 9 C ) < I — t < I — (s — s) cos a. This establishes the theorem for Case II. Q  7  0  Therefore, from the two cases above we have •0(7(s)) < I — (s — s ) c o s a for any a. Hence, ip is a concave function on 7. Now suppose ip(l( )) I some closed interval [a, b]. Let s € [a, b] and let p be a point on dC closest to ( s ) . Since all points on 7|[ ,h] are a distance I from dC, any point on 7I [<,,(,] other than 7(5) is a distance at least I away s  =  o  n  7  a  41  Chapter 4. Proof of The Soul Theorem from p. Let — a be a minimal geodesic from p to ( s ) and let a denote the same geodesic parametrized from ( s ) to p. According to the first variation formula, — a meets j\[ b] orthogonally. Hence a = Z(7'(s), 0^(0)) = f • According to the proof of case I, a = ^ implies that the curves c (t) = exp ( )(s — s)£(t), 0 < t < I have length L[c ] < I. The point c (7) lies on the geodesic r(s) = exp (()(s — s)£(0 and since Z ( ' ( s ) , 0^(0)) = f, then by definition of £ we have Z(£(l), o~' (l)) = ^ . But lemma 4.0.16 implies that since g {v\/L(v, -a«(0) < f } then £(0 0 C \ {o } and hence c {l) g N. Therefore, c (l) must lie on dC or outside of C and since all points on 7|[;,] are at a constant distance I from dC, we must have L[c ] > L Hence L\c ] = I and c (l) G dC. In particular, since the curve c (t) realizes the distance from c (0) to c (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 3  7  3  7  3  a<  s  CT5  t  s  a  CTs  7  s  p  p  s  3  s  a  s  s  s  s  s  F(s,t) = e x p  as(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 |[<,,(,] where V(0) = cr' (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 7  <p(s,t) = e x p  7 ( s )  a  tV{s)  42  Chapter 4. Proof of The Soul Theorem  as shown in Figure 4.5 coincide. dC  V(s) 7(6)  7(5)  7(a)  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 c (t) for s G [a, 6], therefore c' (0) is orthogonal to 7'(s) for all s G [a,b]. Since c (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' (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, II (c' (0),c' (0)) = 0 and s  S  s  s  R  s  s  ds  therefore V i ( 4 ( o ) ) = v | ( c ; ( o ) ) + / ^ ( c ; ( o ) , 4 ( o ) ) = o.  Hence c' (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. • s  We now return to the proof of Theorem 4.1.1. Suppose C = {p G C\d(p,dC) > a} is not totally convex. Then there exists a geodesic 7 : [0,/] —» C with 7(0), (Z) G C but for some s G (0,/), (s) S C . By the definition of C this implies d(~f(s),dC) < a. By the definiton of the distance function ip(t) : C —» K, Proof of Theorem  a  4.1.1.  a  7  -  a  a  7  43  Chapter 4. Proof of The Soul Theorem this is equivalent to the condition that ip(j(s)) < a. But since the endpoints 7(0), 7(7) are contained in C a, we have ^(7(0)) > a and ^>((7)) > Therefore, there exists a minimum of 4>{^{t)) in (0,/). This is a contradiction 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. a  7  C  To prove the second part of the theorem we recall that by definition of we have  m a x  C  max  =  p|  =  C  a  {PeC\d(P,dC)>a}  p|  HCV0) = {p G C\d(p,dC)  =  max}  where max = sup{d(p, dC)\p G C). Therefore, all points in C are equidistant from the boundary of C. Hence, ip is constant on any geodesic 7 in C ..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 and dC it follows that dim C < dim C. • m a x  max  m a x  m a x  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 curvature contains a family of compact totally convex sets Ct associated to each p G M satisfying t2 > h =*• Ch  D Ctl,  Ch  = {q eC \d(q,dCt2) t2  >t  2  -h)  Ut>0Ct = M pedC0  (2) Any closed totally convex set C contains a set N C C which has the structure 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 € C \d(p, 3C ) > a}, 0  C  0  m a x G  = f l c ^ f l C°  Since dC 0 and C is compact, Theorem 4.1.1 implies that C " is totally convex and d i m C ™ < d i m C . For ease of notation, set C ( l ) := C™ *. So C ( l ) is a closed totally convex set with d i m C ( l ) < d i m C . If dC(l) ^ 0 then by applying Theorem 4.1.1 again, this time to C ( l ) , we obtain C(2) := C ( l ) . C(2) is a closed totally convex set and d i m C ( 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 ) 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 i m C ( i ) . B y construction we obtain a sequence of compact totally convex sets C D C ( l ) D C{2) D ••• D C(k). 0  Q  ax  3  0  0  m a x  m a x  a  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 submanifold 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 submanifold without boundary and dimC(k) < d i m C < d i m M . On the other hand, if dimC(fc) = 0 then the set consists of a single point. 0  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 d (q). p  Definition 4.2.1. q ^ p is a critical point of d if for any unit vector u £ T M there exists a distance minimizing geodesic 7 from p to q which satisfies Z(u,-j'(d(p,q))) < f. p  q  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  c=n^) =n(U ^))) c  B  c  t  7  7  t  7  In particular from Proposition 3.0.11, the Ct's expand to give all of M . This implies q £ 0Ct for some t > 0. Since the boundary of C j is nonempty, this set may be contracted by the procedure outlined in the previous section 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 T M 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 C . B y 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 = {veT M\Z(v,y'(0))<%} o  0  0  q  q  a  Now consider -7'(0). Since Z{v,~/'(0)) < § then Z(v, -f{0)) > f and hence — '(0) H. Therefore, — '(0) makes an angle > | with every initial tangent vector to geodesies between q and S. Hence, q is not a critical point of d . • 7  7  s  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| ,c'(0))>| r  for any r £ U and any minimal geodesic c from r to A. gradient-like vectorfieldfor the distance function.  (4.2.4) We call X a  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 vectorfieldfor 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 subset of M and X a gradient-like vectorfieldfor 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 combination 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\ ,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 u = {v G T M\p G S,v ± T S, || V || < e} onto a neighborhood N 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\N 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 <p generated by X. B y 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 r  t  p  p  t  C  t  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 Riemannian 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 R . 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. 3  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 satisfying the properties of a soul. For example, the paraboloid z = x + y in R 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 R with k > 0. 2  2  3  k  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 = x + y in R contains precisely one soul; the vertex. 2  5.2  2  3  The Soul Conjecture  W i t h the establishment of the Soul Theorem, Cheeger and Gromoll [4] proposed 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 nonnegative 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 nonincreasing retraction of the manifold onto the soul. The confirmation of the Soul Conjecture provided a significant breakthrough in the classification of complete noncompact manifolds of nonnegative 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 differential inequality related to those manifolds which statisfy the conditions of the Soul Theorem. In particular, this inequality must be satisfied by a connection, 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 S or M P . 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 manifolds 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 positive curvature. In a related conjecture Hopf claims that although S x S , 2  2  k  l  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 manifolds 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 manifold. 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 resolution 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 curvature 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  Bibliography [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 Geometry, North-Holland Publishing Co., Amsterdam, 1975, North-Holland Mathematical Library, Vol.9 [4] J . Cheeger, D . Gromoll On the structure of complete manifolds of nonnegative curvature, A n n . 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 Manifolds 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, 2004 [8] L . Guijarro, Improving the Metric in an Open Manifold with Nonnegative 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, Transactions of the American Mathematical Society, Vol. 352, No. 1 (2000), 55-59. [10] H . Karcher Riemannian Comparison Constructions, Global Differential Geometry, 170-222, M A A Stud. Math., 27, Math. Assoc. America, Washington, D C , 1989 53  Bibliography [11] J . M . Lee Riemannian Manifolds. An Introduction to Curvature. Graduate Texts in Mathematics, 176. Springer-Velag, New York, 1997 [12] W . Meyer Toponogov's Theorem and Applications, Lecture Notes, College on Differential Geometry, Trieste 1989, notes available at http://wwwmathl.unimuenster .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 Monographs Vol. 149, 1996 [17] K . Tapp Conditions for Nonnegative Curvature on Vector Bundles and Sphere Bundles Duke M a t h Journal. 116, N o . l , (2003), 77-101. [18] J . W . Y i m Space of Souls in a Complete Manifold of Nonnegative Curvature, J . Differential Geometry 32(1992), 429-455.  54  

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