What do you think?
Rate this book
Graduate Texts in Mathematics #94
Foundations of Differentiable Manifolds and Lie Groups
Foundations of Differentiable Manifolds and Lie Groups gives a clear, detailed, and careful development of the basic facts on manifold theory and Lie Groups. Coverage includes differentiable manifolds, tensors and differentiable forms, Lie groups and homogenous spaces, and integration on manifolds. The book also provides a proof of the de Rham theorem via sheaf cohomology theory and develops the local theory of elliptic operators culminating in a proof of the Hodge theorem.
286 pages, Paperback
First published October 10, 1983
4 people are currently reading
62 people want to read
Ratings & Reviews
Friends & Following
Create a free account to discover what your friends think of this book!
Community Reviews
Displaying 1 - 5 of 5 reviews
December 15, 2019
I don't think it's a particularly good book, but it's concise and it takes a slightly different approach to tangent vectors (algebra of germs rather than equivalence classes of curves) so it's a nice reference.
December 16, 2019
It is an introductory book on manifolds, possible reference for the first course on manifolds (first-year grad students).
I only read the last chapter of the book, the Hodge theorem, so my review is limited and based on the last chapter. The chapter is about the Hodge decomposition theorem, some applications and a proof of the theorem.
The Hodge decomposition theorem: Let M be a compact oriented Riemannian manifold, the space of smooth p-forms on M has orthogonal direct sum decomposition to Harmonic p-forms and the image of Laplacian p-forms.
Before proving the theorem, we see a couple of simple applications of the theorem (e.g. Poincare' Duality).
The proof of the theorem, presented in the book, doesn't seem to be very efficient, I suspect that the techniques in this area (used in the proof) have developed since 1971 and probably you can find an easier proof in Evans' PDE.
Overall it seems to be an accurate, well-written introduction to fundamentals of the theory of manifolds.
I only read the last chapter of the book, the Hodge theorem, so my review is limited and based on the last chapter. The chapter is about the Hodge decomposition theorem, some applications and a proof of the theorem.
The Hodge decomposition theorem: Let M be a compact oriented Riemannian manifold, the space of smooth p-forms on M has orthogonal direct sum decomposition to Harmonic p-forms and the image of Laplacian p-forms.
Before proving the theorem, we see a couple of simple applications of the theorem (e.g. Poincare' Duality).
The proof of the theorem, presented in the book, doesn't seem to be very efficient, I suspect that the techniques in this area (used in the proof) have developed since 1971 and probably you can find an easier proof in Evans' PDE.
Overall it seems to be an accurate, well-written introduction to fundamentals of the theory of manifolds.
April 4, 2021
This book... where to start.
It's good, but I would not read it without the context provided by other books on the subject. In particular, I found Lee's Introduction to Smooth Manifolds and Guillemin & Pollack's Differential Topology to be the perfect accompniments. I can't imagine relying on this book alone.
In particular, I know from reading so many confused questions in online forums that defining the tangent space in terms of the germs of a functions is *not* the best introduction to the concept. It's useful, yes, but it's not nearly as intuititive as the umpteen other ways it might be introduced.
Anyway, good book; shouldn't be read in isolation.
It's good, but I would not read it without the context provided by other books on the subject. In particular, I found Lee's Introduction to Smooth Manifolds and Guillemin & Pollack's Differential Topology to be the perfect accompniments. I can't imagine relying on this book alone.
In particular, I know from reading so many confused questions in online forums that defining the tangent space in terms of the germs of a functions is *not* the best introduction to the concept. It's useful, yes, but it's not nearly as intuititive as the umpteen other ways it might be introduced.
Anyway, good book; shouldn't be read in isolation.
Want to read
January 18, 2009If I ever read this, then I will already be a theoretical physicist. That seems unlikely. But I suppose it is within the realm of possibility.
July 24, 2024
This is by far the best differential topology text I have read cover to cover. Lee's book is nice but is far too long, Guillemin and Pollack is very geometric and covers transversality nicely, but it is hardly at the depth of a graduate text, Hirsch goes deeply into the topology of manifolds, but the exposition is quite poor, and Milnor's text is beautiful but far too short. Warner has none of these problems. He is concise, but expounds when it is necessary, his definitions are often coordinate-free but he always reminds the reader how to convert these statements to local ones, and the exercises are often difficult and serve to teach the reader something through their proofs.
The first two chapters are essentially definitions, so they are a great reference while still providing an adequate exposition, especially if one has seen the definitions of manifold, smooth map, tangent space, etc. before in other contexts.
The third chapter on Lie groups is incredible, using the Frobenius theorem of Chapter 1 as a tool to prove many of the bedrock theorems for the theory of Lie groups rapidly and without nasty computations/linear algebraic decompositions. Along with the extensive exercises, this chapter would prepare one to read any heavy-duty text in representation theory of Lie groups and Lie algebras.
Chapter 4 is excellent, though the exposition is very standard and can be found almost anywhere.
Chapter 5 is an excellent first introduction to axiomatic sheaf cohomology in serve of proving the De Rham theorem. Experience with algebraic topology may help, but he truly crosses every t and dots every i hear, so the proof should be easy to follow for any reader.
Chapter 6 is great. The hodge theorem and some easy consequences are presented first, and the remainder of the chapter are dedicated to proving two analytic lemmas. His proofs are fine, though a shorter path can be found in Serge Lang's SL_2(R) or Evans PDE.
The first two chapters are essentially definitions, so they are a great reference while still providing an adequate exposition, especially if one has seen the definitions of manifold, smooth map, tangent space, etc. before in other contexts.
The third chapter on Lie groups is incredible, using the Frobenius theorem of Chapter 1 as a tool to prove many of the bedrock theorems for the theory of Lie groups rapidly and without nasty computations/linear algebraic decompositions. Along with the extensive exercises, this chapter would prepare one to read any heavy-duty text in representation theory of Lie groups and Lie algebras.
Chapter 4 is excellent, though the exposition is very standard and can be found almost anywhere.
Chapter 5 is an excellent first introduction to axiomatic sheaf cohomology in serve of proving the De Rham theorem. Experience with algebraic topology may help, but he truly crosses every t and dots every i hear, so the proof should be easy to follow for any reader.
Chapter 6 is great. The hodge theorem and some easy consequences are presented first, and the remainder of the chapter are dedicated to proving two analytic lemmas. His proofs are fine, though a shorter path can be found in Serge Lang's SL_2(R) or Evans PDE.
Displaying 1 - 5 of 5 reviews