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.
Riemannian Manifolds: An Introduction to Curvature
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Editorial Reviews
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"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
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Most Helpful Customer Reviews
8 of 8 people found the following review helpful:
5.0 out of 5 stars
Nice graduate text.,
This review is from: Riemannian Manifolds: An Introduction to Curvature (Graduate Texts in Mathematics) (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.
14 of 16 people found the following review helpful:
5.0 out of 5 stars
A nice modern treatment.,
This review is from: Riemannian Manifolds: An Introduction to Curvature (Graduate Texts in Mathematics) (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.
7 of 7 people found the following review helpful:
5.0 out of 5 stars
The printing is not up to the standard of the writing,
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This review is from: Riemannian Manifolds: An Introduction to Curvature (Graduate Texts in Mathematics) (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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Inside This Book
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Key Phrases - Statistically Improbable Phrases (SIPs):
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vector bundles, riemannian metrics, rotation angle theorem, curvature endomorphism, total curvature theorem, adapted orthonormal frame, total covariant derivative, circumference theorem, curved polygon, admissible curve, hyperboloid model, speed parametrization, following product rule, radial geodesic, unit speed curve, geodesic ball, first conjugate point, parallel translate, proper variation, minimizing curve, constant curvature metrics, first variation formula, unit speed geodesic, constant sectional curvature, sectional curvatures Key Phrases - Capitalized Phrases (CAPs): (learn more)
Prove Lemma, The Gauss-Bonnet Theorem, Riemannian Geodesics, Review of Tensors, The Model Spaces of Riemannian Geometry, Theorema Egregium, Geometric Interpretation of Curvature, Geodesics of the Model Spaces, Elementary Constructions Associated, Manifolds of Positive Curvature, Prove Proposition, Some Plane Geometry Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
vector bundles, riemannian metrics, rotation angle theorem, curvature endomorphism, total curvature theorem, adapted orthonormal frame, total covariant derivative, circumference theorem, curved polygon, admissible curve, hyperboloid model, speed parametrization, following product rule, radial geodesic, unit speed curve, geodesic ball, first conjugate point, parallel translate, proper variation, minimizing curve, constant curvature metrics, first variation formula, unit speed geodesic, constant sectional curvature, sectional curvatures Key Phrases - Capitalized Phrases (CAPs): (learn more)
Prove Lemma, The Gauss-Bonnet Theorem, Riemannian Geodesics, Review of Tensors, The Model Spaces of Riemannian Geometry, Theorema Egregium, Geometric Interpretation of Curvature, Geodesics of the Model Spaces, Elementary Constructions Associated, Manifolds of Positive Curvature, Prove Proposition, Some Plane Geometry Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
5
books
cite this book:
- Introduction to Smooth Manifolds by John M. Lee in Back Matter
- Engineering Applications of Noncommutative Harmonic Analysis: With Emphasis on Rotation and Motion Groups by Gregory S. Chirikjian in Back Matter
- Hamilton's Ricci Flow (Graduate Studies in Mathematics) by Bennett Chow in Back Matter
- Visualizing Quaternions (The Morgan Kaufmann Series in Interactive 3D Technology) by Andrew J. Hanson in Back Matter
- An Introduction to Differentiable Manifolds and Riemannian Geometry, Revised, Volume 120, Second Edition (Pure and Applied Mathematics) by William M. Boothby in Back Matter
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