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Totally Geodesic Maps Into Manifolds with No Focal Points

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Totally Geodesic Maps Into Manifolds with No Focal Points TOTALLY GEODESIC MAPS INTO MANIFOLDS WITH NO FOCAL POINTS BY JAMES DIBBLE A dissertation submitted to the Graduate School—New Brunswick Rutgers, The State University of New Jersey in partial fulfillment of the requirements for the degree of Doctor of Philosophy Graduate Program in Mathematics Written under the direction of Xiaochun Rong and approved by New Brunswick, New Jersey October 2014 ABSTRACT OF THE DISSERTATION Totally geodesic maps into manifolds with no focal points BY JAMES DIBBLE Dissertation Director: Xiaochun Rong The space of totally geodesic maps in each homotopy class [F] from a compact Riemannian man- ifold M with non-negative Ricci curvature into a complete Riemannian manifold N with no focal points is path-connected. If [F] contains a totally geodesic map, then each map in [F] is energy- minimizing if and only if it is totally geodesic. When N is compact, each map from a product W × M into N is homotopic to a smooth map that’s totally geodesic on the M-fibers. These results generalize the classical theorems of Eells–Sampson and Hartman about manifolds with non-positive sectional curvature and are proved using neither a geometric flow nor the Bochner identity. They can be used to extend to the case of no focal points a number of splitting theorems proved by Cao– Cheeger–Rong about manifolds with non-positive sectional curvature and, in turn, to generalize a theorem of Heintze–Margulis about collapsing. The results actually require only an isometric splitting of the universal covering space of M and other topological properties that, by the Cheeger–Gromoll splitting theorem, hold when M has non- negative Ricci curvature. The flat torus theorem is combined with a theorem about the loop space of a manifold with no conjugate points to show that the space of totally geodesic maps in [F] is path- connected. A center-of-mass method due to Cao–Cheeger–Rong is used to construct a homotopy to a totally geodesic map when M is compact. The asymptotic norm of a Zm-equivariant metric is used to show that the energies of C1 maps in [F] are bounded below by a constant involving the energy ii of an affine surjection from a flat Riemannian torus onto a flat semi-Finsler torus, with equality for a given map if and only if it is totally geodesic. This builds on work of Croke–Fathi. It is also shown that the ratio of convexity radius to injectivity radius can be made arbitrarily small over the class of compact Riemannian manifolds of any fixed dimension at least two. This uses Gulliver’s examples of manifolds with focal points but no conjugate points. iii Acknowledgments I’m happy to thank my adviser, Xiaochun Rong, for his patient guidance and insight, which have contributed to this work in ways large and small. I’m grateful to Hector Sussmann, Feng Luo, and Xiaojun Huang for arranging financial support over the years, to Feng Luo, Zheng-Chao Han, and Christopher Croke for serving on my dissertation committee, and to the Department of Mathemat- ics at Rutgers University for its helpful environment. The Department of Mathematics at Capital Normal University in Beijing graciously hosted me at different points during this research. I’d also like to thank Penny Smith for many interesting mathematical discussions and, moreover, for advice and encouragement that have been invaluable during difficult times in my graduate career. I’m greatly indebted to Jason DeBlois for many interesting and helpful conversations. I would like to thank Christopher Croke for informing me of Kleiner’s counterexample to the flat torus theorem and for other useful suggestions. While working out an early proof of some of these results, I benefitted from conversations with Penny Smith and Armin Schikorra, whose comments, had I better understood them at the time, might have saved me a considerable amount of effort. A key step in that proof used the work of Karuwannapatana–Maneesawarng, which I was introduced to by Stephanie Alexander. I’m certain that the people mentioned here, among many others, have given me far more than I’ve returned to them. I also know that any attempt to catalog all the ways in which I’ve been assisted through this process would prove inadequate. To those who have gone unmentioned, know that I’m grateful for your help. iv Dedication To Sof´ıa, whose love and support have made this possible, and to our children. v Table of Contents Abstract ::::::::::::::::::::::::::::::::::::::::::::: ii Acknowledgments ::::::::::::::::::::::::::::::::::::::: iv Dedication ::::::::::::::::::::::::::::::::::::::::::: v 1. Introduction :::::::::::::::::::::::::::::::::::::::: 1 2. Preliminaries :::::::::::::::::::::::::::::::::::::::: 7 2.1. Algebraic topology . 7 2.2. Riemannian geometry . 10 2.3. Integration . 12 2.4. Convexity . 14 2.5. Geometric radiuses . 17 2.6. Harmonic maps . 30 2.7. Beta and gamma functions . 36 3. Manifolds with no conjugate points ::::::::::::::::::::::::::: 39 3.1. Topology and geometry . 39 3.2. The set of geodesic loops . 46 3.3. The loop map . 52 4. Manifolds with no focal points :::::::::::::::::::::::::::::: 56 4.1. The flat strip theorem and its consequences . 56 4.2. Heat flow methods . 65 vi 5. Commutative diagrams :::::::::::::::::::::::::::::::::: 70 5.1. Manifolds with non-negative Ricci curvature . 70 5.2. Geometric structure . 72 6. Deformation into totally geodesic maps ::::::::::::::::::::::::: 85 6.1. Riemannian center of mass . 85 6.2. Construction of a homotopy . 90 6.3. Generalization to product domains . 99 7. Energy-minimizing versus totally geodesic ::::::::::::::::::::::: 102 7.1. Energy, length, and intersection . 102 7.2. Asymptotic norm of a periodic metric on Zm . 105 7.3. The Finsler torus and affine surjection associated to [F] . 108 7.4. Main inequalities . 114 8. Further questions ::::::::::::::::::::::::::::::::::::: 128 References ::::::::::::::::::::::::::::::::::::::::::: 130 vii 1 Chapter 1 Introduction A Riemannian manifold N has no conjugate points if the exponential map at each point is non- singular and no focal points if the exponential map on the normal bundle of each geodesic is non- singular. When N is complete, these are equivalent to any two points in the universal Riemannian covering space N being joined by a unique geodesic and, respectively, every distance ball in N being strongly convex. Complete manifolds with no conjugate points or no focal points are known to share, to varying degrees, many of the geometric properties of those with non-positive sectional curvature. It almost goes without saying that manifolds with non-positive curvature have been extensively studied, with rigidity results dating back to the celebrated theorem of Gauss–Bonnet. It’s perhaps less well known that the modern study of manifolds with no conjugate or no focal points dates back more than seventy years, with notable contributions by, for instance, Hedlund– Morse [HM] and Hopf [Ho]. The following characterizations, due to O’Sullivan [O’S1], show why these three types of spaces should be related. Theorem 1.1. (O’Sullivan) Let (N;g) be a complete Riemannian manifold. Then the following hold: (a) N has non-positive sectional curvature if and only if, for every geodesic γ : [0;1) ! N and every d2 k k2 ≥ Jacobi field J along γ, dt2 J 0; (b) N has no focal points if and only if, for every geodesic γ : [0;1) ! N and every non-trivial d 2 Jacobi field J along γ that satisfies J(0) = 0, dt kJk > 0; and (c) N has no conjugate points if and only if, for every geodesic γ : [0;1) ! N, every non-trivial Jacobi field J along γ that satisfies J(0) = 0, and every positive time, kJk2 > 0. It follows that complete Riemannian manifolds with non-positive sectional curvature have no focal points and that those with no focal points have no conjugate points. This refines the well-known theorem of Cartan–Hadamard. On the other hand, Gulliver [Gul] constructed examples of complete 2 Riemannian manifolds with positive sectional curvature but no focal points and examples with focal points but no conjugate points. The latter examples may be used to show that, for each m ≥ 2, r(M) inf inj(M) = 0 over the class of compact m-dimensional manifolds, where r and inj are the convexity and injectivity radiuses. This fills in a gap in the literature pointed out by Berger [Ber]. Even more so than in the case of no conjugate points, many of the major results about Rie- mannian manifolds with non-positive sectional curvature generalize to those with no focal points. These include the center theorem [O’S1], flat torus theorem [O’S2], and higher rank rigidity the- orem [Wat], as well as the fact that, in dimensions up to three, all compact manifolds that admit metrics with no focal points also admit metrics with non-positive curvature [IK]. This is because many of the arguments used to prove results about non-positively curved manifolds actually depend only on the convexity of certain sets or functions that holds under the weaker assumption that there are no focal points. One such result is the flat strip theorem, which states that, in a complete and simply connected manifold N with no focal points, any two geodesic lines with finite Hausdorff distance bound a totally geodesic flat strip. This was proved by O’Sullivan [O’S2], using a result of Goto [Got], and independently by Eschenburg [Esc]. The flat strip theorem implies that the set of axes of an isometry is convex. Another is that N has no focal points if and only if, for each y 2 N, d2(·;y) is a strictly convex function [Eb1]. By contrast, Burns showed that the flat strip theorem may fail for manifolds no conjugate points [Burn1], and an unpublished example of Kleiner shows the same for the flat torus theorem [Kle].
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