Most Helpful Customer Reviews
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26 of 26 people found the following review helpful:
5.0 out of 5 stars
The best "first" introduction to smooth manifolds., September 30, 2005
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.
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20 of 20 people found the following review helpful:
5.0 out of 5 stars
A good introduction for serious students, August 13, 2004
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.
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10 of 10 people found the following review helpful:
5.0 out of 5 stars
Excellent, lucid book on manifolds, January 27, 2004
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.
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