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Introduction to Smooth Manifolds: 218 Hardcover – 26 August 2012

by John Lee (Author)
4.5 4.5 out of 5 stars 120 ratings
Edition: 2nd ed. 2013
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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 fewnew 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.

Previous page
  1. ISBN-10
    1441999817
  2. ISBN-13
    978-1441999818
  3. Edition
    2nd ed. 2013
  4. Publisher
    Springer
  5. Publication date
    26 August 2012
  6. Part of series
    Graduate Texts in Mathematics
  7. Language
    English
  8. Dimensions
    15.6 x 3.81 x 23.39 cm
  9. Print length
    724 pages
  10. See all details
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Product description

Review

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 newtopics 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.

Product details

  • Publisher ‏ : ‎ Springer; 2nd ed. 2013 edition (26 August 2012)
  • Language ‏ : ‎ English
  • Hardcover ‏ : ‎ 724 pages
  • ISBN-10 ‏ : ‎ 1441999817
  • ISBN-13 ‏ : ‎ 978-1441999818
  • Dimensions ‏ : ‎ 15.6 x 3.81 x 23.39 cm
  • Customer Reviews:
    4.5 4.5 out of 5 stars 120 ratings

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John M. Lee

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Livro muito bom.
5.0 out of 5 stars Livro com um conteúdo muito bom.
Reviewed in Brazil on 13 April 2024
Verified Purchase
A única reclamação é com respeito aos correios que além de quererem taxar um livro, ainda amassaram a capa.
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Abraham Hernández
5.0 out of 5 stars Perfect
Reviewed in Mexico on 17 February 2024
Verified Purchase
The quality of paper, bending, cover, etc.
are just perfect to me. I have not any complains about it.

The contents are just amazing. I'm using it as one of the main references for my thesis and I recommend it to anyone who is learning about smooth manifolds.

The proofs are clear and detailed, it has tons of examples and exercises, and also is self-contained. It has all the backround you need to fill in the Appendices section. It is a "graduate text" only because it has some advanced prerequisites, but once you have them, the book feels pretty readable and gentle.
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Morpheus
5.0 out of 5 stars It's thicker than I expected.
Reviewed in Canada on 16 April 2021
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Arrived in good condition. This is going to take some time to read, but I can't wait!
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Dr. Wolfgang Klein
5.0 out of 5 stars Lee's books on smooth and Riemannian manifolds
Reviewed in Germany on 27 March 2021
Verified Purchase
are the best I have in my mathematics library. I frequently use them as reference, just recently to prepare my exam in GRT.
One person found this helpful
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Y.M. Wang
5.0 out of 5 stars A leisurely, encyclopedic intro to differential geometry
Reviewed in the United States on 26 June 2019
Verified Purchase
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.
20 people found this helpful
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