Book:John M. Lee/Introduction to Smooth Manifolds

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