Introduction to Smooth Manifolds

From the three books of Lee that I have and had read, this one was the most difficult for me, perhaps it is because it brings so much material, I mean it is a fat book! It has nearly 800 pages. But Lee writes very well, the Topological Manifolds and Riemaniann Manifolds books are just great, the latter is short which makes it easy, the former is just magical. Well, about this book it starts with material about Smooth Manifolds and smooth maps, then talks about tangent vectors, this chapter helped me understand thoroughly the concept of a vector in a curved space (i.e. a manifold) is just a derivative not an arrow and you cannot move them around like in Euclidean space, then on Chapter 4 explains in detail, the concepts of what is a Submersion, Immersion, and Embeddings, later it goes through Whitney Embedding Theorem= A Manifold of dimension n can always be embedded in R^(2n+1), then talks about Lie Groups and a good chapter on Vector Fields, then follows a chapter on Integral curves and Flows, then Vector Bundles etc etc then Tensors, Riemannian Metrics, Differential Forms, Orientations, then we start getting near what I most like of this book, Chapter 16 talks about integration on manifolds and Stoke's Theorem, then comes De Rham Cohomology and finally brings what I most remember this book for, Chapter 18: "The de Rham Theorem", I haven't found another book that goes so deep on this which basically states that the de Rham Groups are isomorphic to Singular Cohomology Theory, after this chapter it comes material on Distributions and Foliations and some more stuff until ending in chapter 22 with Sympletic Manifolds, it also bring 4 Appendices, it is a good book for self study (my case) because Lee explains very well or at least much better than most mathematician authors which write mathematical books although is the third in my list of Lee's books.

There is not much to say about the quality of the text (and the book). As readers mentioned, this is a first-class textbook, and is probably the best for self-study on this subject (Loring Tu's book is also great, but feels a bit "shallow" in terms of examples and exercises). However, my copy already had to be glued back together, even after careful use. [This is a Springer problem, and I will not take any stars away...] Content-wise, it is actually quite similar to Spivak's Comprehensive Intro to Differential Geometry, Vol. 1 -- It's all about manifolds, without specializing on any particular structure. However, it is definitely slower in pace and gentler. Spivak definitely has a way of introducing details that are interesting but difficult and not necessary for a first-timer to go through. And with Spivak, there are still pages I've had to read and re-read just to understand the point, but Lee's text is gentle enough that I usually at least appreciate what he's trying to do, even if I miss the exact reasoning or details on first read. I suppose it's in part because I am reading it after Spivak. Nevertheless, Lee is the rare mathematician who really tries to make sure that students don't get lost. Cynically, for the reason that the text "hand holds" too much, professional differential geometers are unlikely to rate it particularly highly -- mathematicians want texts to be sleek and laconic -- not a word more than what is logically necessary but indifferent to the student seeing the topic for the first time (like any of the three Rudin texts, for example). It's great if a) you have a great professor or b) you happen to be extremely smart. But for most people engaged in self-study, it quickly leads them to give up.Some say this text book is too verbose, compared to Spivak or Tu. However, there is a good deal of material here that the other two texts don't talk about. For example, there is a good deal of homology theory and algebraic topology, culminating with a long, though very comprehensible proof of de Rham's theorem. Moreover, this is the only one of these texts to deal with Stokes' theorem when there are sharp "corners" involved (which is quite often) instead of brushing it under the rug.However, the relative lack of focus does make the text harder to navigate than other introductory texts.

Many books on this topic just list the results and summarize the proofs. Lee actually shows the work, at least for what I needed. The best book I've seen on the topic.

Summary: This book is all you need to master calculus.Pros: Very detail explanation to any concepts. It’s like someone giving a lecture.Cons: Nothing. I might say it includes topics which are less important that can be omitted to make the book more concise.

The book is very well written and has a lot of details and information about the subject I hardly saw in other books about the topic. I'm taking a math degree at the university and can not assist to class as I have to work. This book is great for me to follow the topics that are teaching in on one of my class which has suggested a different book which honestly is not that good for self learning at least for me. I need to learn though to jump some of the topics this book teaches in order to keep peace with my class.

My favorite introduction to this subject remains Spivak, Differential Geometry, volume I. But unfortunately, that book is no longer in print. This probably makes Lee's book the best currently-available text for a first course on smooth manifolds. On the upside, it is carefully written, and contains many excellent problems. On the downside, it often emphasizes arcane fine points instead of focusing on the big picture. My main worry is that many students may fail to see the forest for the trees!

I bought this textbook for a class on differential topology and have found it to be a great introduction to the subject. I think Lee does a great job of motivating the subject, providing helpful examples, and explaining the proofs beyond the level that might be expected for a typical first-year-grad-level textbook.