Book:John M. Lee/Introduction to Smooth Manifolds

John M. Lee: Introduction to Smooth Manifolds

Published $\text {2003}$, Springer: Graduate Texts in Mathematics

ISBN 0-387-95495-3

Subject Matter

• Topology
• Smooth Manifolds

Contents

Preface

$1 \quad$ Smooth Manifolds

Topological Manifolds

Smooth Structures

Examples of Smooth Manifolds

Manifolds with Boundary

Problems

$2 \quad$ Smooth Maps

Smooth Functions and Smooth Maps

Partitions of Unity

Problems

$3 \quad$ Tangent Vectors

Tangent Vectors

The Differential of a Smooth Map

Computations in Coordinates

The Tangent Bundle

Velocity Vectors of Curves

Alternative Definitions of the Tangent Space

Categories and Functors

Problems

$4 \quad$ Submersions, Immersions, and Embeddings

Maps of Constant Rank

Embeddings

Submersions

Smooth Covering Maps

Problems

$5 \quad$ Submanifolds

Embedded Submanifolds

Immersed Submanifolds

Restricting Maps to Submanifolds

The Tangent Space to a Submanifold

Submanifolds with Boundary

Problems

$6 \quad$ Sard's Theorem

Sets of Measure Zero

Sard's Theorem

The Whitney Embedding Theorem

The Whitney Approximation Theorems

Transversality

Problems

$7 \quad$ Lie Groups

Basic Definitions

Lie Group Homomorphisms

Lie Subgroups

Group Actions and Equivariant Maps

Problems

$8 \quad$ Vector Fields

Vector Fields on Manifolds

Vector Fields and Smooth Maps

Lie Brackets

The Lie Algebra of a Lie Group

Problems

$9 \quad$ Integral Curves and Flows

Integral Curves

Flows

Flowouts

Flows and Flowouts on Manifolds with Boundary

Lie Derivatives

Commuting Vector Fields

Time-Dependent Vector Fields

First-Order Partial Differential Equations

Problems

$10 \quad$ Vector Bundles

Vector Bundles

Local and Global Sections of Vector Bundles

Bundle Homomorphisms

Subbundles

Fiber Bundles

Problems

$11 \quad$ The Cotangent Bundle

Covectors

The Differential of a Function

Pullbacks of Covector Fields

Line Integrals

Conservative Covector Fields

Problems

$12 \quad$ Tensors

Multilinear Algebra

Symmetric and Alternating Tensors

Tensors and Tensor Fields on Manifolds

Problems

$13 \quad$ Riemannian Metrics

Riemannian Manifolds

The Riemannian Distance Function

The Tangent–Cotangent Isomorphism

Pseudo-Riemannian Metrics

Problems

$14 \quad$ Differential Forms

The Algebra of Alternating Tensors

Differential Forms on Manifolds

Exterior Derivatives

Problems

$15 \quad$ Orientations

Orientations of Vector Spaces

Orientations of Manifolds

The Riemannian Volume Form

Orientations and Covering Maps

Problems

$16 \quad$ Integration on Manifolds

The Geometry of Volume Measurement

Integration of Differential Forms

Stokes's Theorem

Manifolds with Corners

Integration on Riemannian Manifolds

Densities

Problems

$17 \quad$ De Rham Cohomology

The de Rham Cohomology Groups

Homotopy Invariance

The Mayer–Vietoris Theorem

Degree Theory

Proof of the Mayer–Vietoris Theorem

Problems

$18 \quad$ The de Rham Theorem

Singular Homology

Singular Cohomology

Smooth Singular Homology

The de Rham Theorem

Problems

$19 \quad$ Distributions and Foliations

Distributions and Involutivity

The Frobenius Theorem

Foliations

Lie Subalgebras and Lie Subgroups

Overdetermined Systems of Partial Differential Equations

Problems

$20 \quad$ The Exponential Map

One-Parameter Subgroups and the Exponential Map

The Closed Subgroup Theorem

Infinitesimal Generators of Group Actions

The Lie Correspondence

Normal Subgroups

Problems

$21 \quad$ Quotient Manifolds

Quotients of Manifolds by Group Actions

Covering Manifolds

Homogeneous Spaces

Applications to Lie Theory

Problems

$22 \quad$ Symplectic Manifolds

Symplectic Tensors

Symplectic Structures on Manifolds

The Darboux Theorem

Hamiltonian Vector Fields

Contact Structures

Nonlinear First-Order PDEs

Problems

Appendix $\text{A} \quad$ Review of Topology

Topological Spaces

Subspaces, Products, Disjoint Unions, and Quotients

Connectedness and Compactness

Homotopy and the Fundamental Group

Covering Maps

Appendix $\text{B} \quad$ Review of Linear Algebra

Vector Spaces

Linear Maps

The Determinant

Inner Products and Norms

Direct Products and Direct Sums

Appendix $\text{C} \quad$ Review of Calculus

Total and Partial Derivatives

Multiple Integrals

Sequences and Series of Functions

The Inverse and Implicit Function Theorems

Appendix $\text{D} \quad$ Review of Differential Equations

Existence, Uniqueness, and Smoothness

Simple Solution Techniques

References

Notation Index

Subject Index