# Book:John M. Lee/Introduction to Smooth Manifolds

## John M. Lee: *Introduction to Smooth Manifolds*

Published $2003$, **Springer: Graduate Texts in Mathematics**

- ISBN 0-387-95495-3.

### Subject Matter

### 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