Lectures on Chern-Weil theory and Witten deformations
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 Andr...
| Main Authors: | , |
|---|---|
| Corporate Authors: | , |
| Format: | Electronic Book |
| Language: | English |
| Published: |
River Edge, N.J. :
World Scientific,
c2001
Singapore ; River Edge, N.J. : World Scientific Pub. Co., c2001 |
| Series: | Nankai tracts in mathematics ;
v. 4 |
| Subjects: |
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| 100 | 1 | |a Zhang, Weiping | |
| 100 | 1 | |a Zhang, Weiping, |d 1964- | |
| 245 | 1 | 0 | |a Lectures on Chern-Weil theory and Witten deformations |h [electronic resource] / |c Weiping Zhang |
| 260 | |a River Edge, N.J. : |b World Scientific, |c c2001 | ||
| 260 | |a Singapore ; |a River Edge, N.J. : |b World Scientific Pub. Co., |c c2001 | ||
| 300 | |a xi, 117 p | ||
| 490 | 1 | |a Nankai tracts in mathematics ; |v 4 | |
| 490 | 1 | |a Nankai tracts in mathematics ; |v v. 4 | |
| 504 | |a Includes bibliographical references and index | ||
| 505 | 0 | |a 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 | |
| 506 | |a Access restricted by licensing agreement | ||
| 520 | |a 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 | ||
| 533 | |a Electronic reproduction |b Palo Alto, Calif. : |c ebrary, |d 2009. |n Available via World Wide Web. |n Access may be limited to ebrary affiliated libraries. | ||
| 533 | |a Electronic reproduction |b Singapore : |c World Scientific Publishing Co., |d 2001. |n System requirements: Adobe Acrobat Reader. |n Mode of access: World Wide Web. |n Available to subscribing institutions. | ||
| 590 | |a Access is available to the Yale community | ||
| 650 | 0 | |a Chern classes | |
| 650 | 0 | |a Complexes | |
| 650 | 0 | |a Index theorems | |
| 710 | 2 | |a World Scientific (Firm) | |
| 710 | 2 | |a ebrary, Inc | |
| 776 | 1 | |z 9789810246853 | |
| 776 | 1 | |z 9810246854 | |
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| 830 | 0 | |a Nankai tracts in mathematics ; |v v. 4 | |
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