Riemannian Geometry

Description

"Perhaps it could best be described as an excellent monograph on geodesics which has been expanded to include an introduction to the preliminary ideas which this requires. [...] The main subject of this book is exciting and new results are constantly being obtained. Any differential geometer should know something about geodesics and this is a good place to read about it, as well as to find many useful ideas and examples for reference." Mathematical Reviews (review of the first edition) "Now in its second edition, this textbook continuous to offer the probably best introduction to the topics in Riemannian geometry which are centred around the theory of geodesics." Zentralblatt für Mathematik Chapter 1: Foundations 1.0 Review of Differential Calculus and Topology 1.1 Differentiable Manifolds 1.2 Tensor Bundles 1.3 Immersions and Submersions 1.4 Vector Fields and Tensor Fields 1.5 Covariant Derivation 1.6 The Exponential Mapping 1.7 Lie Groups 1.8 Riemannian Manifolds 1.9 Geodesics and Convex Neighborhoods 1.10 Isometric Immersions 1.11 Riemannian Curvature 1.12 Jacobi Fields Chapter 2: Curvature and Topology 2.1 Completeness and Cut Locus 2.1 Appendix – Orientation 2.2 Symmetric Spaces 2.3 The Hilbert Manifold of H1-curves 2.4 The Loop Space and the Space of Closed Curves 2.5 The Second Order Neighborhood of a Critical Point 2.5 Appendix – The S1- and the Z2-action on AM 2.6 Index and Curvature 2.6 Appendix – The Injectivity Radius for 1/4-pinched Manifolds 2.7 Comparison Theorems for Triangles 2.8 The Sphere Theorem 2.9 Non-compact Manifolds of Positive Curvature Chapter 3: Structure of the Geodesic Flow 3.1 Hamiltonian Systems 3.2 Properties of the Geodesic Flow 3.3 Stable and Unstable Motions 3.4 Geodesics on Surfaces 3.5 Geodesics on the Ellipsoid 3.6 Closed Geodesies on Spheres 3.7 The Theorem of the Three Closed Geodesics 3.8 Manifolds of Non-Positive Curvature 3.9 The Geodesic Flow on Manifolds of Negative Curvature 3.10 The Main Theorem for Surfaces of Genus 0 References Index