Introduction to Smooth Manifolds by Lee

[micromass]

• 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:

[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]

 

[micromass]

This is without a doubt one of the absolute best mathematics books I've ever read. The book covers a lot of smooth manifold theory. Of course, it can't cover everything, so things on Lie groups, curvature, connections are being left out. But Lee really shows a lot of love and passion for the subject. He's an excellent teacher.

One thing I like about the book is that he gives small, easy exercises in the main text which help you reinforce the material. Harder problems are given at the end of each chapter. I wish more math authors would do things like this.

If you liked this book, then be sure to check the sequel: "Riemannian Manifolds: an introduction to curvature".

 

[Sankaku]

Ha! I just got the brand-new 2nd edition (2012) for Christmas. It is superb!

(He has added a bit more material on Lie Groups and Riemannian metrics.)

It is definitely a book aimed a mathematicians, though. If you are physicist just wanting a quick intro to manifolds, then this will probably take you too deep down the rabbit-hole...

 

[WannabeNewton]

Ha! I just got the brand-new 2nd edition (2012) for Christmas. It is superb!

(He has added a bit more material on Lie Groups and Riemannian metrics.)

It is definitely a book aimed a mathematicians, though. If you are physicist just wanting a quick intro to manifolds, then this will probably take you too deep down the rabbit-hole...

Yes but you can never bee too safe. It is easy to misconstrue things and form a false understanding of the underlying mathematical principles for a given field theory if one simply does a cursory overview of differential topology. Plus it's fun =D

 

[Kreizhn]

This is my favourite example of excellent mathematical exposition. It is rigorous but not overly technical, abstract but not unmotivated, thorough but not overly verbose. It can be used as a learning text or a reference text, and is an absolute pleasure to read.

I'm not sure I've ever met a geometer who didn't have a copy of this book, nor have I ever met a geometer whose copy was not stained by coffee, chalk, or had damage to the spine. Let its war wounds be a sign of the endearment with which we hold this book.

 

[WannabeNewton]

This is my favourite example of excellent mathematical exposition. It is rigorous but not overly technical, abstract but not unmotivated, thorough but not overly verbose. It can be used as a learning text or a reference text, and is an absolute pleasure to read.

I'm not sure I've ever met a geometer who didn't have a copy of this book, nor have I ever met a geometer whose copy was not stained by coffee, chalk, or had damage to the spine. Let its war wounds be a sign of the endearment with which we hold this book.

I'm not going to lie, this post was poetic enough to move me. I think I'm going to go to sleep with my copy of the book now xD.

 

[Sankaku]

This is my favourite example of excellent mathematical exposition.

This is slightly off topic, but Lee has a great 6-page PDF on good mathematical writing style. It applies to people just learning to write proofs, all the way to people who are writing textbooks.

http://www.math.washington.edu/~lee/Writing/writing-proofs.pdf

 

[atyy]

Real mathematicians like this book? It's actually understandable even for non-mathematicians. I thought they only like indigestible stuff like Rudin.

 

[micromass]

Real mathematicians like this book? It's actually understandable even for non-mathematicians. I thought they only like indigestible stuff like Rudin.

I know quite some mathematicians who truly dislike Rudin too I guess it's a matter of taste...

 

[atyy]

I know quite some mathematicians who truly dislike Rudin too I guess it's a matter of taste...

I remember one comic in the student newspaper at MIT some years ago. It showed frantic activity in the maths office. Someone had sounded the alert," Quick! Hide all the easy to understand books - an engineer is coming!"

 

[micromass]

I remember one comic in the student newspaper at MIT some years ago. It showed frantic activity in the maths office. Someone had sounded the alert," Quick! Hide all the easy to understand books - an engineer is coming!"

Let's be serious, that never happens http://forums.scifi-nexus.com/images/smilies/nabble/smiley_whistling.gif