Riemannian Manifolds: An Introduction to Curvature (Graduate Texts in Mathematics) | 
| Author: John M. Lee
Publisher: Springer
Category: Book
List Price: $49.95
Buy New: $40.00
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New (20) Used (7) from $36.38
Avg. Customer Rating:  4 reviews
Sales Rank: 238524
Media: Paperback
Edition: 1
Number Of Items: 1
Pages: 252
Shipping Weight (lbs): 0.8
Dimensions (in): 9.1 x 6.1 x 0.6
ISBN: 0387983228
Dewey Decimal Number: 516.373
EAN: 9780387983226
ASIN: 0387983228
Publication Date: September 5, 1997
Availability: Usually ships in 1-2 business days
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Product Description
This text is designed for a one-quarter or one-semester graduate couse in Riemannian geometry. It 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. The book begins with a careful treatment of the machinery of metrics, connections, and geodesics, and then introduces the Riemann curvature tensor, before moving on the submanifold theory, in order to give the curvature tensor a concrete quantitative interpretation. The remainder of the text is devoted to 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. This unique volume will especially appeal to students by presenting a selective introduction to the main ides of the subject in an easily accessible way. The material is ideal for a single course, but broad enough to provide students with a firm foundation from which to pursue research or develop applications in Riemannian geometry and other fields that use its tools. Of special interest are the "exercises" and "problems" dispersed throughout the text. The exercises are carefully chosen and timed so as to give the reader opportunities to review material that hasjust been introduced, to practice working with the definitions, and to develop skills that are used later in the book. The problems that conclude the chapters are generally more difficult. They not only introduce new mateiral not covered in the body of the text, but they also provide the students with indispensable practice in using the
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| Customer Reviews:
 As always September 2, 2007
2 out of 2 found this review helpful
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.
Nice graduate text. March 29, 2007
2 out of 2 found this review helpful
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.
A nice modern treatment. October 26, 2005
7 out of 8 found this review helpful
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.
Excellent reading, even for a layman! October 20, 2005
39 out of 63 found this review helpful
I never had much use for formal education and quit school back in the 10th grade. I work on the line at a fish cannery and do an honest day's work for an honest day's wage. I don't understand people who make a living sitting around all day just thinking or writing things. What's getting made? How do you just think about things and expect people to pay you for it?
Normally I kick back with a cold brew and whatever sports is playing on the tube. Last book I read was in school. I was too busy with football, basketball and girls to waste time with studying. So you might think, what in the world would make me pick up "Riemannian Manifolds" and start reading a graduate text in mathematics? I don't know, something about the title just grabbed me.
You know what? It's a pretty good book. I'm not saying I understood everything Mr. Lee was talking about. I mean, I sorta remember stuff like algebra and geometry and triangles and proofs and things like that, and all that math stuff helped me get through the first four chapters. But when I got to chapter 5, talking about Riemannian geodesics, I got kinda lost. I took a piece of string, used it to connect two cities on a globe, and then I understood. After that, the book picked up pace and finished really strong with comparisons of manifolds on both positive and negative curvatures. I'm thinking I'll read "The Laplacian on a Riemannian Manifold" next. Who ever thought all this math stuff could be so interesting?
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