| Description |
1 online resource (xi, 117 pages). |
| Physical Medium |
polychrome |
| Description |
text file |
| Series |
Nankai tracts in mathematics ; 4
|
|
Nankai tracts in mathematics ; v. 4.
|
| Bibliography |
Includes bibliographical references and index. |
| Contents |
Ch. 1. Chern-Weil theory for characteristic classes. 1.1. Review of the de Rham cohomology theory. 1.2. Connections on vector bundles. 1.3. The curvature of a connection. 1.4. Chern-Weil theorem. 1.5. Characteristic forms, classes and numbers. 1.6. Some examples. 1.7. Bott vanishing theorem for foliations. 1.8. Chern-Weil theory in odd dimension. 1.9. References -- ch. 2. Bott and Duistermaat-Heckman formulas. 2.1. Berline-Vergne localization formula. 2.2. Bott residue formula. 2.3. Duistermaat-Heckman formula. 2.4. Bott's original idea. 2.5. References -- ch. 3. Gauss-Bonnet-Chern theorem. 3.1. A toy model and the Berezin integral. 3.2. Mathai-Quillen's Thom form. 3.3. A transgression formula. 3.4. Proof of the Gauss-Bonnet-Chern theorem. 3.5. Some remarks. 3.6. Chern's original proof. 3.7. References -- ch. 4. Poincaré-Hopf index formula: an analytic proof. 4.1. Review of Hodge theorem. 4.2. Poincaré-Hopf index formula. 4.3. Clifford actions and the Witten deformation. 4.4. An estimate outside of [symbol]. 4.5. Harmonic oscillators on Euclidean spaces. 4.6. A proof of the Poincaré-Hopf index formula. 4.7. Some estimates for [symbol]. 4.8. An alternate analytic proof. 4.9. References -- ch. 5. Morse inequalities: an analytic proof. 5.1. Review of Morse inequalities. 5.2. Witten deformation. 5.3. Hodge theorem for ([symbol]). 5.4. Behaviour of [symbol] near the critical points of f. 5.5. Proof of Morse inequalities. 5.6. Proof of proposition 5.5. 5.7. Some remarks and comments. 5.8. References -- ch. 6. Thom-Smale and Witten complexes. 6.1. The Thorn-Smale complex. 6.2. The de Rham map for Thom-Smale complexes. 6.3. Witten's instanton complex and the map [symbol]. 6.4. The map [symbol]. 6.5. An analytic proof of theorem 6.4. 6.6. References -- ch. 7. Atiyah theorem on Kervaire semi-characteristic. 7.1. Kervaire semi-characteristic. 7.2. Atiyah's original proof. 7.3. A proof via Witten deformation. 7.4. A generic counting formula for k(M). 7.5. Non-multiplicativity of k(M). 7.6. References. |
| Summary |
This invaluable book is based on the notes of a graduate course on differential geometry which the author gave at the Nankai Institute of Mathematics. It consists of two parts: the first part contains an introduction to the geometric theory of characteristic classes due to Shiing-shen Chern and André Weil, as well as a proof of the Gauss-Bonnet-Chern theorem based on the Mathai-Quillen construction of Thom forms; the second part presents analytic proofs of the Poincaré-Hopf index formula, as well as the Morse inequalities based on deformations introduced by Edward Witten. |
| Local Note |
eBooks on EBSCOhost EBSCO eBook Subscription Academic Collection - North America |
| Subject |
Chern classes.
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Chern classes. |
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Index theorems.
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Index theorems. |
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Complexes.
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Complexes. |
| Genre/Form |
Electronic books.
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| Other Form: |
Print version: Zhang, Weiping. Lectures on Chern-Weil theory and Witten deformations. River Edge, N.J. : World Scientific, ©2001 (DLC) 2001046629 |
| ISBN |
9812386580 (electronic book) |
|
9789812386588 (electronic book) |
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9789810246853 |
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9810246854 |
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9810246862 (paperback) |
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