Introduction to Smooth Manifolds (Graduate Texts in Mathematics Book 218) 2nd Edition by John Lee (PDF)

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.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.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This was the assigned textbook for a class I was taking. My introduction to Manifold theory started with Differential Geometry and Topology in undergrad coursework. This book takes it to the next level, building on those foundations and adding new layers of theory and intricacy. Prof Lee is well-known in Differential Geometry circles (and spheres), and his expertise shines through. I thought th book was written clearly, and offered great exercises. The exercises aren’t overwhelming, but they are challenging (which is what you’d want). As someone who loves working with differential forms, I especially liked the later chapters. Another book I’ll be keeping for years to come.PS: I personally don’t have a StackExchange account, but if you do you can often times get hints and tips (but no complete answers!) from Prof Lee himself on the site. His tips & hints are super useful, and you can definitely work your way through any exercise with the bits of info he gives there.

⭐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.

⭐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.

⭐well written

⭐Excellent text book

⭐This is not a review on the contents of the book. The contents are fantastic and Lee is a master expositer. My misgivings lie with the physical quality of the book and with all the Springer books that I have ordered through Amazon. The bindings are poor quality and have a tendency to break even after light use.

⭐A very well written text in a world of poorly texts on this subject. So many authors seem to confound this elegant subject with a profusion of notation that is either poorly defined or garrulously described creating a mess for the student to unravel. Lee has done a superior job. He hasreasonably organized the material and gives proper definitions with a good amount of description. Although a few of the lemmas and theoremscould have a little better motivation and proof, the text is overall one of the best.

⭐This is a really great book. I loved Jack Lee’s intro topology book and was expecting this text to be just as great. I really like his expositions and problems, both of which keep me interested in continuing reading the book.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.

⭐Amazon deve specificare se il libro è stampato da Amazon oppure no.Nel caso in cui il libro sia stato stampato da Amazon la qualità delle pagine è pessima. A me è arrivato un libro stampato da Amazon e non vale la pena pagare così tanto per questa qualità di stampa.Es una grosería que manden estas copias mal empastadas del libro original (el cual es un clásico). Ahórrense el desencanto.Modern THE book introducing smooth manifold theory that every graduate student must read. At a level suitable for graduate student, but covers huge amount of material which might take more than a year to go through.

⭐are the best I have in my mathematics library. I frequently use them as reference, just recently to prepare my exam in GRT.

⭐the hardcover version of this book will fall apart right when you get it (see other reviews). it’s not even glued along the spine of the book. the entire book is held together by two cardboard pages that are glued to the front and back cover. DO NOT GET THIS

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