# Geometry of Differential Forms

American Mathematical Soc., 2001 - Mathematics - 321 pages
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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### Contents

 Manifolds 1 11 What is a manifold? 2 12 Definition and examples of manifolds 11 13 Tangent vectors and tangent spaces 23 14 Vector fields 36 15 Fundamental facts concerning manifolds 44 Differential Forms 57 22 Various operations on differential forms 69
 42 Laplacian and harmonic forms 153 43 The Hodge theorem 158 44 Applications of the Hodge theorem 162 Vector Bundles and Characteristic Classes 169 52 Geodesics and parallel translation of vectors 180 53 Connections in vector bundles and curvature 185 54 Pontrjagin classes 193 55 Chern classes 204

 23 Frobenius theorem 80 24 A few facts 89 The de Rham Theorem 95 31 Homology of manifolds 96 32 Integral of differential forms and the Stokes theorem 104 33 The de Rham theorem 111 34 Proof of the de Rham theorem 119 35 Applications of the de Rham theorem 133 Laplacian and Harmonic Forms 145
 56 Euler classes 211 57 Applications of characteristic classes 216 Fiber Bundles and Characteristic Classes 231 62 𝓢¹ bundles and Euler class 240 63 Connections 257 64 Curvature 265 65 Characteristic classes 275 66 A couple of items 285 Copyright