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

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John M. Lee: Introduction to Riemannian Manifolds

Published $\text {2018}$, Springer: Graduate Texts in Mathematics

ISBN 978-3319917542.


Subject Matter


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