Foundations of Differentiable Manifolds and Lie Groups (Graduate Texts in Mathematics) (v. 94) [Hardcover]

Frank W. Warner (Author)

Book Description

ISBN-10: 0387908943 | ISBN-13: 978-0387908946 | Publication Date: October 10, 1983

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.

Product Details

• Hardcover: 292 pages
• Publisher: Springer (October 10, 1983)
• Language: English
• ISBN-10: 0387908943
• ISBN-13: 978-0387908946
• Product Dimensions: 9.3 x 6.2 x 0.8 inches
• Shipping Weight: 1.6 pounds (View shipping rates and policies)
• Average Customer Review: 3.2 out of 5 stars  See all reviews (6 customer reviews)
• Amazon Best Sellers Rank: #487,939 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

21 of 23 people found the following review helpful:

4.0 out of 5 stars Great book, but not perfect or ideal for every purpose, April 28, 2006

By 

nehiker (Boston) - See all my reviews

This review is from: Foundations of Differentiable Manifolds and Lie Groups (Graduate Texts in Mathematics) (v. 94) (Hardcover)

This is a solid introduction to the foundations (and not just the basics) of differential geometry. The author is rather laconic, and the book requires one to work through it, rather than read it. It presupposes firm grasp of point-set topology, including paracompactness and normality. The basics (Inverse and Implicit Function Theorems, Frobenius Theorem, orientation, and rudiments of de Rham cohomology) are covered in about 100 pages (Chapters 1, 2, and 4). This is not really suitable for an undergraduate course in differential geometry, but is great for a graduate course.

Chapter 3, 5, and 6 (self-contained introductions to Lie Groups, Sheaf Theory, and Hodge Theory, all from a geometric viewpoint) are a really nice feature. The book can't be covered in one semester, but these chapters are great for selft-study. In fact, the organization of Chapter 5 is more suitable for self-study than for being taught in class (lots of theory developed first, with all applications delayed until the end). The real jewel of the book is Chapter 6, a very clean introduction to Hodge Theory, with immediate applications.

The main drawback of the book in my view is that the author avoids vector bundles like the plague. These could have been very nicely incorporated into the book. No mention is made of Mayer-Vietoris or Kunneth formula, even though the former follows easily from the section on cochain complexes in Chapter 5 and the latter with some effort from Chapter 6. There is no mention of manifolds with boundary either, except as regular domains of manifolds for the purpose of Stokes Theorem.

The organization of the book could have been better as well. In particular, the section on cochain complexes could have been incorporated in the rather short de Rham Cohomology Chapter 4, so that MV could have been proved and used to compute the cohomology of spheres (beyond the circle). Some subsections, including in Chapter 1, appear out of order to me. There is a shortage of exercises in my view. Some of the author's notation (for tangent spaces, tangent bundles) is rather non-standard.

However, all-in-all, I can't think of a better differential geometry text for a graduate course. Spivak and Lee are quite wordy and do not have the same breadth. Either book would be preferable to Warner for an undergraduate course though. The price is a relative bargain too.

24 of 28 people found the following review helpful:

4.0 out of 5 stars A good book if you have some background, November 25, 2001

By 

G.B.S "17" (Europe) - See all my reviews

This review is from: Foundations of Differentiable Manifolds and Lie Groups (Graduate Texts in Mathematics) (v. 94) (Hardcover)

This book is a good introduction to manifolds and lie groups.
Still if you dont have any background ,this is not the book to start with.The first chapter is about the basics of manifolds:vector fields,lie brackts,flows on manifolds and more,
this chapter can help one alot as a second book on the subject.
The second chapter is about tensors, this introduction can be very hard for someone who didnt met the notion of tensors ,since the book tends to take a very general line with out down to earth examples.the 3ed chapter is about lie groups.It is avery good introduction ,to my point of view ,one of the best there is.
The 4th chapter is about integration on manifolds and is very good too.Chapters 5and6 are about De Rham cohomology theory and the hodge theorem.
If you have some knowledge on all the above subjects this book can serve as a very good overview on the subject.

8 of 11 people found the following review helpful:

4.0 out of 5 stars Good, as long as you have enough background, November 21, 2003

By A Customer

This review is from: Foundations of Differentiable Manifolds and Lie Groups (Graduate Texts in Mathematics) (v. 94) (Hardcover)

I read this book at the very beginning of my studying in differential geometry and was striked. The definitions and methods used in this book seemed totally incomprehensible to me. However, after some development in this field, I found that this book is very concise. It is a very good surey on differential geometry but not a good book to start with. Definitions are given from the most "down to bottom" one. It is a very good attitude, yet, if you do not have much background in differential geometry, this book may takes you several days in order to understand the concept of tensor and exterior algebra.