Riemannian Geometry

Wilhelm P.A. Klingenberg

• Publisher: De Gruyter
• Year: 2011
• ISBN:
  9783110905120
• Edition: 2
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

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Table of contents

• 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