A Course in Differential Geometry

by
Format: Hardcover
Pub. Date: 2000-10-01
Publisher(s): Amer Mathematical Society
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Summary

This textbook for second-year graduate students is intended as an introduction to differential geometry with principal emphasis on Riemannian geometry. Chapter I explains basic definitions and gives the proofs of the important theorems of Whitney and Sard. Chapter II deals with vector fields and differential forms. Chapter III addresses integration of vector fields and p-plane fields. Chapter IV develops the notion of connection on a Riemannian manifold considered as a means to define parallel transport on the manifold. The author also discusses related notions of torsion and curvature, and gives a working knowledge of the covariant derivative. Chapter V specializes on Riemannian manifolds by deducing global properties from local properties of curvature, the final goal being to determine the manifold completely. Chapter VI explores some problems in PDEs suggested by the geometry of manifolds. The author is well-known for his significant contributions to the field of geometry and PDEs - particularly for his work on the Yamabe problem - and for his expository accounts on the subject. The text contains many problems and solutions, permitting the reader to apply the theorems and to see concrete developments of the abstract theory.

Table of Contents

Background material Differentiable manifolds
Tangent space Integration of vector fields and differential forms
Linear connections Riemannian manifolds
The Yamabe problem-An introduction to research
Bibliography
Subject index Notation
Table of Contents provided by Publisher. All Rights Reserved.

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