Geometry of Differential Forms

Front Cover
American Mathematical Soc., 2001 - Mathematics - 321 pages
2 Reviews
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
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Intégrales singulières
Frédéric Pham
No preview available - 2005

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