- #1
- 22,183
- 3,324
- Author: John Lee
- Title: Introduction to Smooth Manifolds
- Amazon link https://www.amazon.com/dp/0387954481/?tag=pfamazon01-20
- Prerequisities: Topology, Linear algebra, Calculus 3. Some analysis wouldn't hurt either.
- Level: Grad
Table of Contents:
Code:
[LIST]
[*] Smooth Manifolds
[LIST]
[*] Topological Manifolds
[*] Smooth Structures
[*] Examples of Smooth Manifolds
[*] Manifolds with Boundary
[*] Problems
[/LIST]
[*] Smooth Maps
[LIST]
[*] Smooth Functions and Smooth Maps
[*] Partitions of Unity
[*] Problems
[/LIST]
[*] Tangent Vectors
[LIST]
[*] 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
[/LIST]
[*] Submersions, Immersions, and Embeddings
[LIST]
[*] Maps of Constant Rank
[*] Embeddings
[*] Submersions
[*] Smooth Covering Maps
[*] Problems
[/LIST]
[*] Submanifolds
[LIST]
[*] Embedded Submanifolds
[*] Immersed Submanifolds
[*] Restricting Maps to Submanifolds
[*] The Tangent Space to a Submanifold
[*] Submanifolds with Boundary
[*] Problems
[/LIST]
[*] Sard’s Theorem
[LIST]
[*] Sets of Measure Zero
[*] Sard’s Theorem
[*] The Whitney Embedding Theorem
[*] The Whitney Approximation Theorems
[*] Transversality
[*] Problems
[/LIST]
[*] Lie Groups
[LIST]
[*] Basic Definitions
[*] Lie Group Homomorphisms
[*] Lie Subgroups
[*] Group Actions and Equivariant Maps
Problems
[/LIST]
[*] Vector Fields
[LIST]
[*] Vector Fields on Manifolds
[*] Vector Fields and Smooth Maps
[*] Lie Brackets
[*] The Lie Algebra of a Lie Group
[*] Problems
[/LIST]
[*] Integral Curves and Flows
[LIST]
[*] Integral Curves
[*] Flows
[*] Flowouts
[*] Flows and Flowouts on Manifolds with Boundary
[*] LieDerivatives
[*] Commuting Vector Fields
[*] Time-Dependent Vector Fields
[*] First-Order Partial Differential Equations
[*] Problems
[/LIST]
[*] Vector Bundles
[LIST]
[*] Vector Bundles
[*] Local and Global Sections of Vector Bundles
[*] Bundle Homomorphisms
[*] Subbundles
[*] Fiber Bundles
[*] Problems
[/LIST]
[*] The Cotangent Bundle
[LIST]
[*] Covectors
[*] The Differential of a Function
[*] Pullbacks of Covector Fields
[*] Line Integrals
[*] Conservative Covector Fields
[*] Problems
[/LIST]
[*] Tensors
[LIST]
[*] Multilinear Algebra
[*] Symmetric and Alternating Tensors
[*] Tensors and Tensor Fields on Manifolds
[*] Problems
[/LIST]
[*] Riemannian Metrics
[LIST]
[*] Riemannian Manifolds
[*] The Riemannian Distance Function
[*] The Tangent–Cotangent Isomorphism
[*] Pseudo-Riemannian Metrics
[*] Problems
[/LIST]
[*] Differential Forms
[LIST]
[*] The Algebra of Alternating Tensors
[*] Differential Forms on Manifolds
[*] Exterior Derivatives
[*] Problems
[/LIST]
[*] Orientations
[LIST]
[*] Orientations of Vector Spaces
[*] Orientations of Manifolds
[*] The Riemannian Volume Form
[*] Orientations and Covering Maps
[*] Problems
[/LIST]
[*] Integration on Manifolds
[LIST]
[*] The Geometry of Volume Measurement
[*] Integration of Differential Forms
[*] Stokes’s Theorem
[*] Manifolds with Corners
[*] Integration on Riemannian Manifolds
[*] Densities
[*] Problems
[/LIST]
[*] De Rham Cohomology
[LIST]
[*] The de Rham Cohomology Groups
[*] Homotopy Invariance
[*] The Mayer–Vietoris Theorem
[*] Degree Theory
[*] Proof of the Mayer–Vietoris Theorem
[*] Problems
[/LIST]
[*] The de Rham Theorem
[LIST]
[*] Singular Homology
[*] Singular Cohomology
[*] Smooth Singular Homology
[*] The de Rham Theorem
[*] Problems
[/LIST]
[*] Distributions and Foliations
[LIST]
[*] Distributions and Involutivity
[*] The Frobenius Theorem
[*] Foliations
[*] Lie Subalgebras and Lie Subgroups
[*] Overdetermined Systems of Partial Differential Equations
[*] Problems
[/LIST]
[*] The Exponential Map
[LIST]
[*] One-Parameter Subgroups and the Exponential Map
[*] The Closed Subgroup Theorem
[*] Infinitesimal Generators of Group Actions
[*] The Lie Correspondence
[*] Normal Subgroups
[*] Problems
[/LIST]
[*] Quotient Manifolds
[LIST]
[*] Quotients of Manifolds by Group Actions
[*] Covering Manifolds
[*] Homogeneous Spaces
[*] Applications to Lie Theory
[*] Problems
[/LIST]
[*] Symplectic Manifolds
[LIST]
[*] Symplectic Tensors
[*] Symplectic Structures on Manifolds
[*] The Darboux Theorem
[*] Hamiltonian Vector Fields
[*] Contact Structures
[*] Nonlinear First-Order PDEs
[*] Problems
[/LIST]
[*] Appendix: Review of Topology
[LIST]
[*] Topological Spaces
[*] Subspaces, Products, Disjoint Unions, and Quotients
[*] Connectedness and Compactness
[*] Homotopy and the Fundamental Group
[*] Covering Maps
[/LIST]
[*] Appendix: Review of Linear Algebra
[LIST]
[*] Vector Spaces
[*] Linear Maps
[*] The Determinant
[*] Inner Products and Norms
[*] Direct Products and Direct Sums
[/LIST]
[*] Appendix: Review of Calculus
[LIST]
[*] Total and Partial Derivatives
[*] Multiple Integrals
[*] Sequences and Series of Functions
[*] The Inverse and Implicit Function Theorems
[/LIST]
[*] Appendix: Review of Differential Equations
[LIST]
[*] Existence, Uniqueness, and Smoothness
[*] Simple Solution Techniques
[/LIST]
[*] References
[*] Notation Index
[*] Subject Index
[/LIST]
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