Riemannian Geometry

Name: Riemannian Geometry
Price: 89.82 USD
Availability: InStock
ISBN: 9780387292465

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…

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