Book:John M. Lee/Introduction to Riemannian Manifolds/Second Edition

Subject Matter

 * Definition:Differential Geometry
 * Riemannian Manifolds

Contents

 * Preface


 * $1 \quad$ What Is Curvature?
 * The Euclidean Plane
 * Surfaces in Space
 * Curvature in Higher Dimensions


 * $2 \quad$ Riemannian Metrics
 * Definitions
 * Methods for Constructing Riemannian Metrics
 * Basic Constructions on Riemannian Manifolds
 * Lengths and Distances
 * Pseudo-Riemannian Metrics
 * Other Generalizations of Riemannian Metrics
 * Problems


 * $3 \quad$ Model Riemannian Manifolds
 * Symmetries of Riemannian Manifolds
 * Euclidean Spaces
 * Spheres
 * Hyperbolic Spaces
 * Invariant Metrics on Lie Groups
 * Other Homogeneous Riemannian Manifolds
 * Model Pseudo-Riemannian Manifolds
 * Problems


 * $4 \quad$ Connections
 * The Problem of Differentiating Vector Fields
 * Connections
 * Covariant Derivatives of Tensor Fields
 * Vector and Tensor Fields Along Curves
 * Geodesics
 * Parallel Transport
 * Pullback Connections
 * Problems


 * $5 \quad$ The Levi-Civita Connection
 * The Tangential Connection Revisited
 * Connections on Abstract Riemannian Manifolds
 * The Exponential Map
 * Normal Neighborhoods and Normal Coordinates
 * Tubular Neighborhoods and Fermi Coordinates
 * Geodesics of the Model Spaces
 * Euclidean and Non-Euclidean Geometries
 * Problems


 * $6 \quad$ Geodesics and Distance
 * Geodesics and Minimizing Curves
 * Uniformly Normal Neighborhoods
 * Completeness
 * Distance Functions
 * Semigeodesic Coordinates
 * Problems


 * $7 \quad$ Curvature
 * Local Invariants
 * The Curvature Tensor
 * Flat Manifolds
 * Symmetries of the Curvature Tensor
 * The Ricci Identities
 * Ricci and Scalar Curvatures
 * The Weyl Tensor
 * Curvatures of Conformally Related Metrics
 * Problems


 * $8 \quad$ Riemannian Submanifolds
 * The Second Fundamental Form
 * Hypersurfaces
 * Hypersurfaces in Euclidean Space
 * Sectional Curvatures
 * Problems


 * $9 \quad$ The Gauss–Bonnet Theorem
 * Some Plane Geometry
 * The Gauss–Bonnet Formula
 * The Gauss–Bonnet Theorem
 * Problems


 * $10 \quad$ Jacobi Fields
 * The Jacobi Equation
 * Basic Computations with Jacobi Fields
 * Conjugate Points
 * The Second Variation Formula
 * Cut Points
 * Problems


 * $11 \quad$ Comparison Theory
 * Jacobi Fields, Hessians, and Riccati Equations
 * Comparisons Based on Sectional Curvature
 * Comparisons Based on Ricci Curvature
 * Problems


 * $12 \quad$ Curvature and Topology
 * Manifolds of Constant Curvature
 * Manifolds of Nonpositive Curvature
 * Manifolds of Positive Curvature
 * Problems


 * Appendix $\text{A} \quad$ Review of Smooth Manifolds
 * Topological Preliminaries
 * Smooth Manifolds and Smooth Maps
 * Tangent Vectors
 * Submanifolds
 * Vector Bundles
 * The Tangent Bundle and Vector Fields
 * Smooth Covering Maps


 * Appendix $\text{B} \quad$ Review of Tensors
 * Tensors on a Vector Space
 * Tensor Bundles and Tensor Fields
 * Differential Forms and Integration
 * Densities


 * Appendix $\text{C} \quad$ Review of Lie Groups
 * Definitions and Properties
 * The Lie Algebra of a Lie Group
 * Group Actions on Manifolds


 * References


 * Notation Index


 * Subject Index