# 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

### 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