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The Structure of Manifolds of Nonnegative Sectional Curvature

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The Structure of Manifolds of Nonnegative Sectional Curvature The Structure of Manifolds of Nonnegative Sectional Curvature by Christy Cameron B.Sc.(Hon.), Dalhousie University, 2005 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE 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 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 ii Table of Contents iii 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 iii 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 (Rn) 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 VM on M and the induced connection V5 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.
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