Customer Reviews

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

5 star:		(5)
4 star:		(2)
3 star:		(1)
2 star:		(0)
1 star:		(0)

 

 

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.

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.

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.

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.

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.

This book is great. An excellent book for anyone with a little background in mathematical formality, i.e. analysis, topology or differential geometry.

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?

I'm not qualified to give the review, but this is one of the top books in the quantum systems engineering group that John Sidles recommends and reads. Plus, anything with Riemannian in the title recalls Cryptonomicon which can't be all bad.