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Riemannian Manifolds: An Introduction to Curvature (Graduate Texts in Mathematics) [Paperback]

John M. Lee
4.6 out of 5 stars  See all reviews (9 customer reviews)

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Book Description

September 23, 1997 0387983228 978-0387983226 1997

This text focuses on developing an intimate acquaintance with the geometric meaning of curvature and thereby introduces and demonstrates all the main technical tools needed for a more advanced course on Riemannian manifolds. It covers proving the four most fundamental theorems relating curvature and topology: the Gauss-Bonnet Theorem, the Cartan-Hadamard Theorem, Bonnet’s Theorem, and a special case of the Cartan-Ambrose-Hicks Theorem.


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Riemannian Manifolds: An Introduction to Curvature (Graduate Texts in Mathematics) + Introduction to Smooth Manifolds (Graduate Texts in Mathematics, Vol. 218) + Introduction to Topological Manifolds (Graduate Texts in Mathematics)
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Editorial Reviews

Review

"This book is very well writen, pleasant to read, with many good illustrations. It deals with the core of the subject, nothing more and nothing less. Simply a recommendation for anyone who wants to teach or learn about the Riemannian geometry."
Nieuw Archief voor Wiskunde, September 2000

Product Details

  • Series: Graduate Texts in Mathematics (Book 176)
  • Paperback: 226 pages
  • Publisher: Springer; 1997 edition (September 23, 1997)
  • Language: English
  • ISBN-10: 0387983228
  • ISBN-13: 978-0387983226
  • Product Dimensions: 9.1 x 6 x 0.5 inches
  • Shipping Weight: 12 ounces (View shipping rates and policies)
  • Average Customer Review: 4.6 out of 5 stars  See all reviews (9 customer reviews)
  • Amazon Best Sellers Rank: #883,877 in Books (See Top 100 in Books)

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Customer Reviews

4.6 out of 5 stars
(9)
4.6 out of 5 stars
Most Helpful Customer Reviews
18 of 20 people found the following review helpful
5.0 out of 5 stars The printing is not up to the standard of the writing January 9, 2009
Format:Paperback
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.
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9 of 9 people found the following review helpful
3.0 out of 5 stars Do Carmo's is better May 12, 2010
Format:Paperback
I've taught an introductory differential geometry course from Lee's book, and in retrospect Do Carmo's "Riemannian Geometry" would have been a better choice. To be fair Lee does masterful job introducing basic concepts from curvature to Jacobi fields, but here are a few things I disliked. The book assumes working knowledge of smooth manifolds and Lie brackets, while many students need review of the former, and know nothing of the latter. Lee doesn't give enough examples beyond constant curvature spaces: there is virtually no mention of warped products, Riemannian submersions, Lie groups, or homogeneous spaces. Exercises are few, unmotivated, and their difficulty is in stark contrast with the easiness of the main text. I feel Do Carmo's book is superior in all respects, and last time I checked it was not much more expensive.
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9 of 9 people found the following review helpful
5.0 out of 5 stars Nice graduate text. March 29, 2007
Format:Paperback
I used this book to teach about half a year of a graduate Riemannian manifolds course. It is a very good introductory text. I wish it has a bit more background on curves and surfaces, but otherwise it was excellent. It doesn't get into a lot of more advanced topics, but the treatment of Jacobi fields and so forth is really good.
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16 of 19 people found the following review helpful
5.0 out of 5 stars A nice modern treatment. October 26, 2005
Format:Hardcover
I just got this fella, and I'm really just through the first four chaptors but so far I'm very pleased. He really tries to tie the definitions and theorems to something you can think about. He gives three "model spaces", the n-sphere, R^n, and hyperbolic space and keeps coming beck to them as he does new things. I like that after he defines connections he shows some in R^n. You know, things like that. Anyway, I'm not a specialist but this seems to me as good an introduction to Reimannian curvature as you could ask for. At least as good in my opinion as Del Carmo's book.

So thanks again Dr. Lee. You keep writing them and we'll keep reading them.
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8 of 9 people found the following review helpful
4.0 out of 5 stars As always September 2, 2007
Format:Paperback
Prof. Lee sets the norm of mathematical exposition. I would give it 5 stars if it were more comprehensive. There is so much to say about Riemannian manifolds and it would be a pleasure to see them under the light the author sheds on such subtle concepts. One very nice feature of the book that underlies its structure is that it uses four theorems - pillars of Riemannian geometry as a guide of what should be included. This approach, besides improving considerably the organization of the book as compared to other books on the subject, it also motivates the reader who now has a target in his mind, namely the proofs of these important theorems. It is really nontrivial to introduce people to mathematical areas as broad as Riemannian geometry. Also, useful suggestions are given in the preface for further reading.
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