Introduction to Smooth Manifolds (Graduate Texts in Mathematics) Hardcover – 26 Aug 2012
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From the reviews of the second edition:
“It starts off with five chapters covering basics on smooth manifolds up to submersions, immersions, embeddings, and of course submanifolds. … the book under review is laden with excellent exercises that significantly further the reader’s understanding of the material, and Lee takes great pains to motivate everything well all the way through … . a fine graduate-level text for differential geometers as well as people like me, fellow travelers who always wish they knew more about such a beautiful subject.” (Michael Berg, MAA Reviews, October, 2012)
From the Back Cover
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.
This second edition has been extensively revised and clarified, and the topics have been substantially rearranged. The book now introduces the two most important analytic tools, the rank theorem and the fundamental theorem on flows, much earlier so that they can be used throughout the book. A few new topics have been added, notably Sard’s theorem and transversality, a proof that infinitesimal Lie group actions generate global group actions, a more thorough study of first-order partial differential equations, a brief treatment of degree theory for smooth maps between compact manifolds, and an introduction to contact structures.
Prerequisites include a solid acquaintance with general topology, the fundamental group, and covering spaces, as well as basic undergraduate linear algebra and real analysis.See all Product Description
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Most Helpful Customer Reviews on Amazon.com (beta)
I imagine someone with experience in differential geometry might find Lee's style tedious. However, as Lee himself points out in the introduction, students encountering differential geometry for the first time usually need this level of detail. I wholeheartedly agree with him. Differential geometry is a difficult subject to get into, with lots and lots of notation and a tendency to handwave technical details. I personally found Lee's book far superior to the relatively sparse class notes provided by my instructor. A careful reading of Lee's book is slow going, but there's no question that you will understand the material after you're done. (I wish the same could be said for all math books...)
A nice feature is the very frequent exercises interspersed in the text. These delegate the easier or more repetitive proofs to the reader--and they're designed to be easy. This gives the reader a chance to get comfortable with the mechanics of the subject before being expected to tackle the more difficult problem sets at the end of each chapter. Too often math courses skip this phase of the learning process.
A few remarks about the many details provided in the text: I think they only add pedagogical value, most, if not all, of which can be glossed over by more experienced readers. Some of these details are significant (even by way of the embedded "exercises", not so much the end-of-chapter "problems" -- the author makes the distinction between the two), and don't disrupt the flow of the text in a detrimental way. And though some of these exercises are routine, they're great for checking understanding, and give the reader some confidence. And sometimes it's just nice to see some things spelled out for you when you're working through the details (of a technical proof) yourself. In other words, I don't feel that the author gets bogged down too much with the finer points so as to lose sight of what's really important. Though this makes for a bulky text, I'm actually quite fond of bulky math books myself! To the student, especially a beginner, these can be instructive.
Despite all this, I still think text is quite an enjoyable read -- even pleasantly chatty in places -- and serves as a solid introduction to smooth manifolds in particular, and differential geometry in general.
If you haven't read his other book on topology, I suggest doing so first. This book definitely feels like the "next" book, or a sequel to his topology book, if you will.