Customer Reviews

Introduction to Smooth Manifolds

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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.

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.

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.

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!

that lends itself to self-studying then this is not a good book, but excellent. All complaints reported in other reviews are actually answered in the preface: the book is about the mathematical machinery ordinated under the title smooth manifold theory. It is not a book on Riemannian geometry that's why there is no extensive treatment of metrics or any treatment of connections. Each topic comes up whenever the prerequisite tools have been built and enough motivation can be given, that's why it is a pleasure to read this book. If you like encyclopedic expositions there are plenty of them out there. The appendix contains a compilation of virtually all the results you need from calculus, linear algebra, and topology in order to study smooth manifolds. Perhaps the first time I actually read the appendix of a book (Arnol'd 's books form an exception) It is obvious that the author belongs to that group of people who like to excel in whatever they do, as all his books not only teach you the subject of their titles but also how to write a book if it happens to reach that point in your mathematical career. They are in some sense both books and meta-books on mathematics.
This review is not intended to comment on other reviews, but let us be honest and agree on the fact that an author never faces the danger of being too clear: as to the length and the pace of the book, I wish this book were only one volume of a series from the same author starting with topology and culminating with the interplay of differential geometry and PDE. There is a drawback however, reasonably not anticipated. Most math books are not written to be actually read (aphoristic but true). This one makes an exception and thus the binding proves insufficient quickly. A hardcover version would be convenient. Suggestion for clever math students: learn the stuff from Lee and then pretend you are reading Lang's "introduction"...

I should say first that I was already familiar with manifold theory before picking up this book. I had already wrestled with some of the definitions, theorems, and whatnot, so I can't necesarily say I was a complete beginner before reading this book. Also, I'm not sure if I can say how great this book would be if you have no idea what a manifold (or tangent space, etc.) is. However, that stuff aside, this is an amazing text. I'm studying this book on my own, and it's great. The concepts are woven throughout the text instead of being lumped into chapters devoted to them (though some people might prefer the latter). Also, they're used to reinforce and build on each other.

As an example, Spivak doesn't treat Lie groups until the second to last chapter. Lee introduces them in the second chapter, uses them as examples throughout the text, builds up the theory of Lie groups as the book goes on, uses Lie groups (and their actions on other manifolds) in developing certain other areas (it really streamlines the development) and ends with a nice big chapter on them. Of course, this is just one example.

Lee developes manifold theory so that it would appeal to a physicist, geometer, algebraist, topologist, etc. Everything gets talked about! This means, however, that he can't treat any one subject in too much detail. For instance, he leaves curvature and other parts of Riemannian geometry to his other Riemannian Geometry text, but it's definitely worth the trade off. This book trashes Spivak. Buy it today!

I have taught from John Lee's book and do not share the enthusiasm of other reviewers. My main objection is that his proofs tend to be stodgy and labored. I usually find it easier and more satisfying to prove the theorems myself rather than read his proofs. Consider, for example, his proof to the inverse function theorem. It is four pages long, but it does not need to be. His writing is rarely concise and the reader has to work too hard to pull the main points out of the myriad of details. Lee resorts to coordinate proofs at every turn, when more elegant coordinate free approaches are possible. It is nice that most students are able to follow his proofs because he leaves out nothing. But they are not always able to see the forest for the trees. Lee omits the proof to Sard's Theorem and the Whitney Extension Theorem, two basic results in this area. Thus he cannot cover degree theory. His discussion of the pushforward of a vector field is awkward. He inappropriately uses pushforward notation for the total derivative linear mapping approximating a smooth mapping near a point. On a positive note, the book is an excellent resource for the study of Lie Groups and Lie Algebras. Milnor's little blue book is one of the most elegant texts in differential topology, but it does not cover as much. Spivak is excellent, as is Hirsch.

It's very readable. He has a good descriptive, conversational style. It's also very thorough. For example after he gives his definitions of the tangent space he copmares and it to the competitors and shows equivalence. There is plenty of work in coordinates but things are defined in the proper coordinate invariant ways. Nice coverage of vector bundles and a whole chaptor on the cotangent bundle which is nice.

Lots of Lie groups... he introduces symplectic manifolds and talks about Hamiltonian mechanics on the cotangent bundle. What I'm saying is all and all he talks about a lot of wicked good stuff.

One warning: The word transversality appears I believe once in the whole book and that's in an exercise. Intersection theory does not seem to be covered at all. That's not a complaint. That stuff is in lots of good books that don't go anywhere near a lot of the things that are in Lee's book. I'm just saying if you are thinking of using this as a reference for a course that has transversality on the syllabus you will need a second book. Let's say Hirsch's differential topology for the classic, or Guillemin and Pollack's book by the same name for something that doesn't have function spaces as it's second chapter.

So yeah. Good book. Thanks Dr. Lee.

I just finished a 20-week course from this book. It is well-written, with a healthy number of examples and many exercises (interspersed throughout the text) and problems (at the end of each chapter). The style is rather informal: this is good for the novice to this subject, which groans under the weight of its own notation. The presentation is well-organized, clear, and accessible. Dr. Lee maintains current errata for the book (some did make it into the problems unfortunately) at his website.

One thing I might suggest is that if you plan to use this book heavily (e.g., for a course rather than for reference or bedtime reading) you should consider investing in the hardcover version is possible. The book is lengthy and the binding tends to split. Mine is still in one piece, but only just. You have to be very gentle with this book to keep it intact.

By all accounts, this and Dr. Lee's other two books on manifolds are exceptionally well-written. But my copies arrived from Amazon this week, and, unfortunately, Amazon and Springer have decided to replace the crisp offset-printing of earlier printings by lower quality digitally-printed versions, probably as a cost-cutting measure.

If you care about how books look, I'd suggest trying Amazon marketplace or small retailers elsewhere to increase your odds of getting a superior copy from an earlier printing.