Riemannian Manifolds: An Introduction to Curvature (Graduate Texts in Mathematics) 1997th Edition by John M. Lee (PDF)

Description

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.

User’s Reviews

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 About the Author John M. Lee has been a mathematics professor at the University of Washington in Seattle since 1987. He has written two other popular graduate texts (Introduction to Smooth Manifolds and Introduction to Topological Manifolds), and an undergraduate text (Axiomatic Geometry).

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

⭐All books by Lee deserve 5 stars that is to start saying. This book objective is tp introduce Curvature in Riemannian Manifolds. For this, he starts introducing Vectors, Tensors, Manifolds and Vector Bundles. The he defines a metric and the a connection and prooves the existence a torsionless symmetryc conecction whic is the Levi-Civita connection. Afterwards he talks about Covariant Differentiation and Geodesics to finanlly give the Curvature Tensor of Riemann. You see, in Physics, the curvature tensor is presented or derived as the commutator of covariant derivatives but that path leaves out one term which when working with coordinated basis is zero but not always! Therefore this book goes on to show an identity betweem the commutator of covariant derivatives and the Lie bracket which is valid only on flat spaces, therefore, any departure from this identity is what mathematicians call the curvature tensor. In this way the missing term in Physics is no longer missing and there appears as a covariant derivative of a vector with respect to a Lie bracket, very amusing! The second feature that I highlight this book for is that it gives the explanation and enunciation of a generalization of the Gauss-Bonnet theorem but this time for any dimensions due to the great mathematician Chern. All in all, a very good book that I definitely recommend for anyone interested in geometry of manifolds.

⭐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.

⭐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.

⭐A very nice exposition on the material. The presentation is nice and a bit different from a regular text book I think. The exercises are nicely placed after the appropriate discussion and not just bunched at the end of a chapter. The chapters are small and nicely arranged. I think if you have had a course in Differential Geometry already, this book will be a good idea to reinforce the concepts and give you a proper flavor of Riemannian Geometry.

⭐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.

⭐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’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.

⭐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 bought this textbook for the sole purpose to learn about the geometric interpretation of curvature in higher dimensions. As I had expected, the geometric interpretation for the Ricci and scalar curvatures was covered in chapter 8 “Riemannian Submanifolds”. These curvatures play an important role in the theory of general relativity and their geometric interpretations are often sought after, as the general relativity textbooks normally don’t delve into this matter. You need to know the fundamentals of differential manifolds to take on this textbook.

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