Geometry to Differential Forms

Geometry to Differential Forms

by Shigeyuki Morita
     
 

ISBN-10: 0821810456

ISBN-13: 9780821810453

Pub. Date: 09/01/2001

Publisher: American Mathematical Society

Since the times of Gauss, Riemann, and Poincare, one of the principal goals of the study of manifolds has been to relate local analytic properties of a manifold with its global topological properties. Among the high points on this route are the Gauss-Bonnet formula, the de Rham complex, and the Hodge theorem; these results show, in particular, that the central tool

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Overview

Since the times of Gauss, Riemann, and Poincare, one of the principal goals of the study of manifolds has been to relate local analytic properties of a manifold with its global topological properties. Among the high points on this route are the Gauss-Bonnet formula, the de Rham complex, and the Hodge theorem; these results show, in particular, that the central tool in reaching the main goal of global analysis is the theory of differential forms. This book is a comprehensive introduction to differential forms. It begins with a quick presentation of the notion of differentiable manifolds and then develops basic properties of differential forms as well as fundamental results about them, such as the de Rham and Frobenius theorems. The second half of the book is devoted to more advanced material, including Laplacians and harmonic forms on manifolds, the concepts of vector bundles and fiber bundles, and the theory of characteristic classes. Among the less traditional topics treated in the book is a detailed description of the Chern-Weil theory. With minimal prerequisites, the book can serve as a textbook for an advanced undergraduate or a graduate course in differential geometry.

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Product Details

ISBN-13:
9780821810453
Publisher:
American Mathematical Society
Publication date:
09/01/2001
Series:
Translations of Mathematical Monographs, #201
Edition description:
New Edition
Pages:
321
Product dimensions:
5.50(w) x 8.40(h) x 0.80(d)

Table of Contents

Preface
Preface to the English Edition
Outline and Goal of the Theory
Ch. 1Manifolds1
Ch. 2Differential Forms57
Ch. 3The de Rham Theorem95
Ch. 4Laplacian and Harmonic Forms145
Ch. 5Vector Bundles and Characteristic Classes169
Ch. 6Fiber Bundles and Characteristic Classes231
References315
Index317

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