Introduction to Smooth Manifolds | 
 enlarge | Author: John M. Lee
Publisher: Springer
Category: Book
List Price: $59.95
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Rating:  14 reviews
Sales Rank: 16436
Media: Paperback
Edition: 1
Pages: 648
Number Of Items: 1
Shipping Weight (lbs): 1.6
Dimensions (in): 9.2 x 6.1 x 1.2
ISBN: 0387954481
Dewey Decimal Number: 514.3
EAN: 9780387954486
Publication Date: September 23, 2002
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Product Description
This book is an introductory graduate-level textbook on the theory of smooth manifolds. Its goal is to familiarize students with the tools they will need in order to use manifolds in mathematical or scientific research--- smooth structures, tangent vectors and covectors, vector bundles, immersed and embedded submanifolds, tensors, differential forms, de Rham cohomology, vector fields, flows, foliations, Lie derivatives, Lie groups, Lie algebras, and more. The approach is as concrete as possible, with pictures and intuitive discussions of how one should think geometrically about the abstract concepts, while making full use of the powerful tools that modern mathematics has to offer. Along the way, the book introduces students to some of the most important examples of geometric structures that manifolds can carry, such as Riemannian metrics, symplectic structures, and foliations. The book is aimed at students who already have a solid acquaintance with general topology, the fundamental group, and covering spaces, as well as basic undergraduate linear algebra and real analysis. John M. Lee is Professor of Mathematics at the University of Washington in Seattle, where he regularly teaches graduate courses on the topology and geometry of manifolds. He was the recipient of the American Mathematical Society's Centennial Research Fellowship and he is the author of two previous Springer books, Introduction to Topological Manifolds (2000) and Riemannian Manifolds: An Introduction to Curvature (1997).
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| Customer Reviews: Read 9 more reviews...
A good introduction for serious students August 13, 2004
Guy Lebanon (West Lafayette, IN USA)
17 out of 17 found this review helpful
This book is a very nice introduction to smooth (differentiable manifolds). Explanations are lucid, the style is consistent, and there is a feeling of a real textbook (not just a collection of results).
However, note that the book assumes quite a bit of previous knowledge. Readers are assumed to be familiar with basic topology (a standard assumption in such books), and with some algebraic topology. As a preparation for this book, I recommend the first several chapters of the author's book "Introduction to Topological Manifolds".
Readers that have this background, or that are willing to learn it elsewhere will benefit from this book greatly. I strongly recommend this book over Boothby and Spivak. It is more advanced and contains more content; in addition, it is also clearer and more pleasant to read.
Several disadvantages:
There are several typos in the book, including some that are not found in the author's errata on his web site. It is not clear what chapters are necessary for understanding later chapters. For example, if you want to skip Lie groups and algebras, will you still understand the chapter on tensors?
The biggest disadvantage is that important material such as curvature and connections is missing. For that material you need to buy the author's earlier book "Riemannian manifolds: an introduction to curvature". I think that at least some treatment should have been included for people who will not get that additional book.
With all this in mind, this is an EXCELLENT book. I have tried several other books on the topic and this is the winner by a big margin.
The best "first" introduction to smooth manifolds. September 30, 2005
Sean Raleigh (San Diego, CA)
19 out of 19 found this review helpful
The problem with most differential geometry books out there--even the ones labeled as "introductory" or "elementary"--is that they are really not that elementary. Authors often assume that once they've introduced a definition or theorem, the reader surely will have mastered it right away. Then, the concept is quickly used to derive other results while the reader is left in the dust.
Lee avoids this common pitfall by dwelling a bit longer on each new idea, often using several examples to give the reader some much-needed practice and exposure to new concepts. It's easy to be a bit lazy when writing books like this and give very few actual computations: authors tend either to use stock examples which are often trivial, or relegate more interesting computations to the exercises--exercises which are basically impossible to do for the first-time reader since they haven't seen a solid example. Lee is committed to helping the reader work through the computations and then gives exercises that follow right behind and are genuinely doable.
A good example of this is the introduction of tangent vectors and vector fields. All books explain these ideas and give the rules for working with them, but Lee takes several chapters to develop the concepts carefully and show how to work with them. The intrinsic definition of a tangent vector is by no means a self-evident or intuitive concept, but you wouldn't know it from looking at other books.
Lee depends on lot on working in coordinates. This is a GOOD thing for first-time readers. Sure, the coordinate-free formulas are cleaner, but they give the reader absolutely no idea how to use them. The disadvantage of working in coordinates is usually the notation, namely the proliferation of indices, but Lee adeptly handles notational complexity by carefully introducing and explaining his notational conventions. And then he sticks to them! Eventually the book does move into more coordinate-free notation, but only after the reader has had a chance to absorb the concepts. Again, Lee takes great care in warning and re-warning the reader when he switches to shortcut notation.
It is precisely this attention to detail and slower pace that causes some to devalue this text. If you are already familiar with the basics of differential geometry and smooth manifold theory, you're probably going to find the pace of this book a bit on the slow side. It is about 600 pages long and the last chapter only manages to get to Lie groups and Lie algebras. (By the way, this last chapter is one of the best things about this book.) Some argue that far more should be accomplished in 600 pages, but I disagree. I read this book cover to cover when I needed to learn differential geometry and I came out understanding what I needed to begin pursing my research topic for my dissertation. I had tried several other "standards" in vain.
There is one more feature that could be perceived as negative. Lee scatters a lot of things around throughout the chapters instead of grouping everything in self-contained chapters. Pedagogically, I think this is the best way to do it if you're reading the book cover to cover or using it in a course. But if you're using it as a resource for looking things up, it can be a bit difficult to find what you want. Fortunately, the index is good and it usually isn't too much trouble.
Overall, highly recommended as a "first" read. Although there are some definite prerequisites for starting this book (namely, a good topology course), any grad student should be able to pick it up and start understanding it right away.
A must-have text for any grad student! February 10, 2007
anonymous
4 out of 6 found this review helpful
We're using Gullemin and Pollack's text for our differential topology course. I found it rather difficult to learn from it. A friend of mine strongly recommended this book by Lee (actually, he recommended the whole series.) The definitions are concrete, and the proofs are rigorous. Lee provides some great motivations for the ideas presented in this text. Ultimately, I find that it's a well written topology book and should be on any mathematicians bookshelf.
Excellent, lucid book on manifolds January 28, 2004
Paul (BERKELEY, CA USA)
9 out of 9 found this review helpful
Topics are explained with exceptional clarity; portions of the book are well tied together; and the order of exposition flows very well. Lie groups are introduced quite early on, but their full power is not revealed until later in the book. I can't laud this book enough. I had a firm, well-developed basis of differential geometry after reading through this book for a course. The excersises are illuminating, as are the examples. Theorems and their proofs are clearly labeled. The motivational explanations prefacing theorems do an excellent job of conveying the intuition behind ideas.I would recommend this book over Boothby any day. I haven't read Spivak, so I can't compare Lee to it, but Lee definitely seemed like an excellent choice for an intro grad class on differential geometry.
A completely solid introduction with great style July 24, 2004
C. Asplund (Santa Barbara, CA, USA)
8 out of 8 found this review helpful
The book has been invaluable to me over the past few months while learning differential topology and geometry. Lee is a careful writer and a gifted teacher, and I'm glad he wrote this introduction to such an important but oft difficult-to-penetrate field. The range of topics covered is extensive and well-organized, including excellent chapters on smooth maps, tangent, cotangent, and vector bundles, Lie group actions, and the best introduction to tensors and differential forms I've encountered. I am anxious to read on to the later chapters.
I hope he continues to write text books!
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