Riemannian Geometry

ISBN-10: 0387292462

ISBN-13: 9780387292465

Edition: 2nd 2006 (Revised)

Authors: Peter Petersen
List price: $64.95
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Description: 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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Book details

List price: $64.95
Edition: 2nd
Copyright year: 2006
Publisher: Springer
Publication date: 8/9/2006
Binding: Hardcover
Pages: 405
Size: 6.25" wide x 9.25" long x 1.00" tall
Weight: 1.628
Language: English

Preface
Riemannian Metrics
Riemannian Manifolds and Maps
Groups and Riemannian Manifolds
Local Representations of Metrics
Doubly Warped Products
Exercises
Curvature
Connections
The Connection in Local Coordinates
Curvature
The Fundamental Curvature Equations
The Equations of Riemannian Geometry
Some Tensor Concepts
Further Study
Exercises
Examples
Computational Simplifications
Warped Products
Hyperbolic Space
Metrics on Lie Groups
Riemannian Submersions
Further Study
Exercises
Hypersurfaces
The Gauss Map
Existence of Hypersurfaces
The Gauss-Bonnet Theorem
Further Study
Exercises
Geodesies and Distance
Mixed Partials
Geodesies
The Metric Structure of a Riemannian Manifold
First Variation of Energy
The Exponential Map
Why Short Geodesies Are Segments
Local Geometry in Constant Curvature
Completeness
Characterization of Segments
Riemannian Isometries
Further Study
Exercises
Sectional Curvature Comparison I
The Connection Along Curves
Second Variation of Energy
Nonpositive Sectional Curvature
Positive Curvature
Basic Comparison Estimates
More on Positive Curvature
Further Study
Exercises
The Bochner Technique
Killing Fields
Hodge Theory
Harmonic Forms
Clifford Multiplication on Forms
The Curvature Tensor
Further Study
Exercises
Symmetric Spaces and Holonomy
Symmetric Spaces
Examples of Symmetric Spaces
Holonomy
Curvature and Holonomy
Further Study
Exercises
Ricci Curvature Comparison
Volume Comparison
Fundamental Groups and Ricci Curvature
Manifolds of Nonnegative Ricci Curvature
Further Study
Exercises
Convergence
Gromov-Hausdorff Convergence
Holder Spaces and Schauder Estimates
Norms and Convergence of Manifolds
Geometric Applications
Harmonic Norms and Ricci curvature
Further Study
Exercises
Sectional Curvature Comparison II
Critical Point Theory
Distance Comparison
Sphere Theorems
The Soul Theorem
Finiteness of Betti Numbers
Homotopy Finiteness
Further Study
Exercises
De Rham Cohomology
Lie Derivatives
Elementary Properties
Integration of Forms
Cech Cohomology
De Rham Cohomology
Poincare Duality
Degree Theory
Further Study
Bibliography
Index
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