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

John M. Lee: Introduction to Riemannian Manifolds

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

ISBN 978-3319917542

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