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Dynamics and Global Geometry of Manifolds Without Conjugate Points

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Dynamics and Global Geometry of Manifolds Without Conjugate Points SOCIEDADE BRASILEIRA DE MATEMÁTICA ENSAIOS MATEMATICOS´ 2007, Volume 12, 1{181 Dynamics and global geometry of manifolds without conjugate points Rafael O. Ruggiero Abstract. Manifolds with no conjugate points are natural generaliza- tions of manifolds with nonpositive sectional curvatures. They have in common the fact that geodesics are global minimizers, a variational prop- erty of geodesics that is quite special. The restriction on the sign of the sectional curvatures of the manifold leads to a deep knowledge about the topology and the global geometry of the manifold, like the characterization of higher rank, nonpositively curved spaces as symmetric spaces. However, if we drop the assumptions concerning the local geometry of the manifold the study of geodesics becomes much harder. The purpose of this survey is to give an overview of the classical theory of manifolds without conju- gate points where no assumptions are made on the sign of the sectional curvatures, since the famous work of Morse about minimizing geodesics of surfaces and the works of Hopf about tori without conjugate points. We shall show important classical and recent applications of many tools of Riemannian geometry, topological dynamics, geometric group theory and topology to study the geodesic flow of manifolds without conjugate points and its connections with the global geometry of the manifold. Such appli- cations roughly show that manifolds without conjugate points are in many respects close to manifolds with nonpositive curvature from the topological point of view. Contents Introduction 7 1 Preliminaries on Riemannian geometry and geodesics 11 1 Riemannian manifolds as metric spaces . 11 2 The Levi-Civita connection . 12 3 The differential equation of geodesics . 13 4 The exponential map, curvature and Jacobi fields . 14 5 The geodesic flow . 17 5.1 Sasaki metric, Jacobi fields and the differential of the geodesic flow . 18 5.2 Transversal symplectic structure of the geodesic flow 20 6 The universal covering . 24 6.1 Homotopy classes . 25 6.2 Covering transformations and fundamental group . 26 6.3 Covering transformations as isometries . 27 2 Dynamical systems and stability 28 1 Limit set and non-wandering set . 28 2 Hyperbolic sets and Anosov dynamics . 30 3 Local product structure, expansiveness and stability . 34 4 The stability conjecture(s) . 45 3 C1 tools to study geometry and dynamics: Jacobi fields, hyperbolicity and global geometry 48 1 Negative curvature and Anosov dynamics . 49 1.1 Rauch's comparison theorem . 49 1.2 Asymptotic Jacobi fields and Riccati equation . 50 1.3 Green subbundles . 53 1.4 Proof of Anosov's theorem and the impact of hyper- bolicity in the global geometry of geodesics . 54 2 Characterization of Anosov geodesic flows in manifolds with- out conjugate points . 55 CONTENTS 2.1 On the divergence of Jacobi fields in manifolds with- out conjugate points . 56 2.2 Quasi-Anosov systems . 58 2.3 Quasi-convexity of Jacobi fields and continuity of Green subbundles . 59 2.4 Hyperbolic behavior of Green subbundles . 60 3 Anosov dynamics and globally minimizing properties of geodesics: hyperbolicity implies the absence of conjugate points . 63 3.1 Conclusions . 65 4 Further applications of the Riccati equation in ergodic theory 66 4 C0 tools to study global geometry of geodesics in the uni- versal covering of manifolds without conjugate points 69 1 Horospheres and Busemann flows in M~ . 70 2 Two-dimensional tori without conjugate points are flat . 75 3 Surfaces without conjugate points of higher genus, quasi- geodesics and quasi-convexity . 77 4 Divergence of geodesic rays and continuity of Hθ(0) with respect to θ . 84 4.1 Asymptoticity and quasi convexity . 84 4.2 Divergence of geodesic rays in surfaces without con- jugate points . 86 4.3 Divergence of geodesic rays in higher dimensions . 89 4.4 The continuity of horospheres . 90 5 Summary of results about the geodesic flow of compact sur- faces without conjugate points . 91 6 Higher dimensional manifolds without conjugate points: Con- vexity and flats in the universal covering . 93 5 Expansive geodesic flows in manifolds without conjugate points: an example of the application of C0 methods to study dynamics and global geometry 95 1 Expansiveness and no conjugate points . 96 2 Expansive geodesic flows without conjugate points in higher dimensions . 97 2.1 Expansiveness implies quasi-convexity and asymp- toticity . 98 2.2 Strong stable and unstable sets . 100 3 Divergence of geodesic rays in M~ . 104 4 Proof of Theorem 5.2 and applications . 104 CONTENTS 6 The fundamental group, global geometry and geometric group theory 108 1 Fundamental group, algebra and geometry . 108 2 Fundamental group in negative curvature, Preissmann's the- orem and Gromov hyperbolic groups . 113 3 Preissmann property implies Gromov hyperbolicity? . 117 4 Expansive dynamics, weak stability and Gromov hyperbolicity120 7 Weak hyperbolicity of the geodesic flow and global hyper- bolic geometry in nonpositive curvature 128 1 Homoclinic geodesics in the two torus and instability . 131 2 The closing lemma for flat planes in analytic manifolds with nonpositive curvature and the proof of the conjecture in this case . 134 3 Sharpening the shadowing result by means of Aubry-Mather theory . 135 8 Weak hyperbolicity of geodesics, global geometry and Poincar´e conjecture 141 1 Homoclinic geodesics in manifolds without conjugate points and variable curvature sign . 143 1.1 A generalization of Hedlund's tunnels . 144 1.2 On finding homoclinic, minimizing geodesics in per- turbed metrics . 146 1.3 Morse homoclinic geodesics and bounded asymptote . 155 2 Bounded asymptote and Preissmann's property . 157 2.1 Shadowing of geodesics and the Preissmann property in bounded asymptote . 158 2.2 Topological stability and Preissmann property . 159 3 Further results in certain manifolds without conjugate points . 160 3.1 Morse homoclinic geodesics in perturbations of met- rics without conjugate points . 161 3.2 On the nonexistence of Morse homoclinic geodesics in manifolds without conjugate points . 162 3.3 Proof of Theorem 8.2 . 164 3.4 Proof of Theorem 8.3 . 165 4 Weak stability and Thurston's geometrization conjecture . 166 4.1 The Poincar´e conjecture . 167 4.2 Thurston's geometrization conjecture . 167 4.3 Applications of the Thurston's geometrization con- jecture to weak stability problems . 169 CONTENTS 5 Further remarks and questions . 173 Bibliography 175 Introduction The theory of manifolds without conjugate points is one of the most chal- lenging research areas in geometry. A complete Riemannian manifold (M; g), where M is a C1 manifold and g is a C1 Riemannian metric in M, has no conjugate points if the exponential map at every point is non-singular. This is equivalent to the fact that every geodesic is glob- ally minimizing in the universal covering M~ endowed with the pullback of the metric g by the covering map. Namely, the distance between any two points in a geodesic in M~ is just the length of the subset of this geodesic bounded by these two points. Hyperbolic space and manifolds of nonposi- tive curvature are well known examples of such manifolds, but the question of knowing if a manifold without conjugate point admits a metric with non- positive curvature is very difficult and open. The absence of conjugate points is on the one hand, a strong condition from the topological point of view; but on the other hand it gives no hints about the local geometry of the manifold. So the usual way in geometry to get the global description of manifolds from local data - like in the theory of manifolds whose sectional curvatures have constant sign - does not pro- ceed in this theory. The geometric theory of manifolds without conjugate points is therefore the result of an exciting interplay between many areas in mathematics, from classical Riemannian geometry and dynamical systems to calculus of variations, classical mechanics and geometric group theory. The purpose of this survey is to give an overview of the main techniques and results about one of the main subjects of the theory: the relationships between the topological dynamics of the geodesic flow, the topology and the global geometry of the manifold. We shall focus on the aspects of the theory which are not part of the theory of manifolds with nonpositive curva- ture, where there are already fairly good, complete surveys and books (see for instance [34], [8], [5], [60]). The theory of manifolds with nonpositive curvature is much richer in results than the theory of manifolds without conjugate points, simply because the convexity of many geometric func- tions and objects determined by the nonpositive curvature is a powerful tool to study global geometry, topology and the dynamics of the geodesic 7 8 Rafael O. Ruggiero flow of the manifold. This is perhaps the reason why the theory of man- ifolds without conjugate points is not so popular: it is much harder and the results are quite often weaker than in the theory of manifolds with nonpositive curvature. The topics presented in the survey could be classified in two main sub- jects: first of all the application of smooth methods or C 1 methods to study the geometry and the dynamics of the geodesic flow in manifolds without conjugate points; and secondly the application of C 0 or variational methods to study geodesics and global geometry in such manifolds. The application of C1 methods concerns of course Jacobi fields and curvature, and such methods allow to study global structures from local data, like in the the- ory of manifolds with nonpositive curvature. We shall illustrate some of these methods, unfortunately we have to choose and many interesting ap- plications won't be considered. The variational or C 0 methods concern the use of the globally minimizing nature of geodesics in the universal covering regardless of the sign of sectional curvatures.
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