This volume introduces techniques and theorems of Riemannian geometry, and opens the way to advanced topics. The text combines the geometric parts of Riemannian geometry with analytic aspects of the theory, and reviews recent research. The updated second edition includes a new coordinate-free formula that is easily remembered (the Koszul formula in disguise); an expanded number of coordinate calculations of connection and curvature; general fomulas for curvature on Lie Groups and submersions; variational calculus integrated into the text, allowing for an early treatment of the Sphere theorem using a forgotten proof by Berger; recent results regarding manifolds with positive curvature.
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P. Petersen Riemannian Geometry "A nice introduction to Riemannian geometry, containing basic theory as well as several advanced topics." —EUROPEAN MATHEMATICAL SOCIETY
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Intended for a one year course, this volume serves as a single source, introducing students to the important techniques and theorems, while also containing enough background on advanced topics to appeal to those students wishing to specialize in Riemannian geometry. This is one of the few works to combine both the geometric parts of Riemannian geometry and the analytic aspects of the theory, while also presenting the most up-to-date research. This book will appeal to readers with a knowledge of standard manifold theory, including such topics as tensors and Stokes theorem. Various exercises are scattered throughout the text, helping motivate readers to deepen their understanding of the subject. Important additions to this new edition include: * A completely new coordinate free formula that is easily remembered, and is, in fact, the Koszul formula in disguise; * An increased number of coordinate calculations of connection and curvature; * General fomulas for curvature on Lie Groups and submersions; * Variational calculus has been integrated into the text, which allows for an early treatment of the Sphere theorem using a forgottten proof by Berger; * Several recent results about manifolds with positive curvature. From reviews of the first edition: "The book can be highly recommended to all mathematicians who want to get a more profound idea about the most interesting achievements in Riemannian geometry. It is one of the few comprehensive sources of this type." - Bernd Wegner, Zentralblatt
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Fundamental mathematics,
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This review is from: Riemannian Geometry (Graduate Texts in Mathematics) (Hardcover)
Dear Sirs
It is only true that Bernard Riemann was one the greatest mathematicians that lived, his works were fundamental for mathematics, vector and mainly tensor calculus that would be the main base of relativity, perhaps he was a mathematician better then Eistein, but he didn't have the fundamental creative genius of Einstein, he developed tensor calculus as well as Kronecker, Levi-Civita, Ricci, his works are there for of the most importance for understang relativity, calculus and mathematics, this book as donated by me to the Lisbon "Tecnico", is a clear book in the purpose of enlightening Riemann mathematics. P. Rose/ M.Lapa
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nonnegative curvature operator, doubly warped products, parallel curvature tensor, soul theorem, hinge version, smooth distance function, relative volume comparison, many diffeomorphism types, rotationally symmetric metrics, variational field, spaces and holonomy, many homotopy types, upper curvature bound, positive curvature operator, nonnegative sectional curvature, curvature pinching, second variation formula, positively curved manifolds, symmetry rank, irreducible symmetric spaces, dual coframe, shape operator, injectivity radius, connectedness principle, curvature bounds Key Phrases - Capitalized Phrases (CAPs): (learn more)
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nonnegative curvature operator, doubly warped products, parallel curvature tensor, soul theorem, hinge version, smooth distance function, relative volume comparison, many diffeomorphism types, rotationally symmetric metrics, variational field, spaces and holonomy, many homotopy types, upper curvature bound, positive curvature operator, nonnegative sectional curvature, curvature pinching, second variation formula, positively curved manifolds, symmetry rank, irreducible symmetric spaces, dual coframe, shape operator, injectivity radius, connectedness principle, curvature bounds Key Phrases - Capitalized Phrases (CAPs): (learn more)
Hodge Laplacian Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
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100
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cite this book:
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- Computers, Rigidity, and Moduli: The Large-Scale Fractal Geometry of Riemannian Moduli Space (Porter Lectures) by Shmuel Weinberger on page 111, and page 158
- Riemannian Geometry (Translations of Mathematical Monographs) by Takashi Sakai in Back Matter (1), and Back Matter (2)
- Metric Structures in Differential Geometry by Gerard Walschap in Back Matter (1), and Back Matter (2)
- Riemann Surfaces (Graduate Texts in Mathematics) (v. 71) by Hershel M. Farkas in Back Matter
- Commutative Algebra: with a View Toward Algebraic Geometry (Graduate Texts in Mathematics) by David Eisenbud in Back Matter
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