Screen LibraryThe index theorem and the heat equation method /
This book provides a self-contained representation of the local version of the Atiyah-Singer index theorem. It contains proofs of the Hodge theorem, the local index theorems for the Dirac operator and some first order geometric elliptic operators by using the heat equation method. The proofs are up...
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| Main Authors: | |
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| Corporate Authors: | |
| Published: |
World Scientific Pub. Co.,
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| Publisher Address: | Singapore : |
| Publication Dates: | 2001. |
| Literature type: | eBook |
| Language: | English |
| Series: |
Nankai tracts in mathematics ;
v. 2 |
| Subjects: | |
| Online Access: |
http://www.worldscientific.com/worldscibooks/10.1142/4691#t=toc |
| Summary: |
This book provides a self-contained representation of the local version of the Atiyah-Singer index theorem. It contains proofs of the Hodge theorem, the local index theorems for the Dirac operator and some first order geometric elliptic operators by using the heat equation method. The proofs are up to the standard of pure mathematics. In addition, a Chern root algorithm is introduced for proving the local index theorems, and it seems to be as efficient as other methods. |
| Carrier Form: | 1 online resource (xix,287pages) : illustrations. |
| Bibliography: | Includes bibliographical references (pages 279-282) and index. |
| ISBN: | 9789812810106 (electronic bk.) |
| Index Number: | QA614 |
| CLC: | O189.3 |
| Contents: | ch. 1. Preliminaries in Riemannian geometry. 1.1. Basic notions of Riemannian geometry. 1.2. Computations by using orthonormal moving frame. 1.3. Differential forms and orthonormal moving frame method. 1.4. Classical geometric operators. 1.5. Normal coordinates. 1.6. Computations on sphere. 1.7. Connections on vector bundles and principal bundles. 1.8. General tensor calculus -- ch. 2. Schr dinger and heat operators. 2.1. Fundamental solution and Levi iteration. 2.2. Existence of fundamental solution. 2.3. Cauchy problem of heat equation. 2.4. Hodge theorem. 2.5. Applications of Hodge theorem. 2.6. Index problem -- ch. 3. MP parametrix and applications. 3.1. MP parametrix. 3.2. Existence of initial solutions. 3.3. Asymptotic expansion for heat kernel. 3.4. Local index for elliptic pperators -- ch. 4. Chern-Weil theory. 4.1. Characteristic forms and characteristic classes. 4.2. General characteristic forms. 4.3. Chern root algorithm. 4.4. Formal approach to local index of signature operator -- ch. 5. Clifford algebra and super algebra. 5.1. Clifford algebra. 5.2. Super algebra. 5.3. Computations on supertraces -- ch. 6. Dirac operator. 6.1. Spin structure. 6.2. Spinor bundle. 6.3. Dirac operator. 6.4. Index of Dirac operator -- ch. 7. Local index theorems. 7.1. Local index theorem for Dirac operator. 7.2. Local index theorem for signature operator. 7.3. Local index theorem for de Rham-Hodge operator -- ch. 8. Riemann-Roch theorem. 8.1. Hermitian metric. 8.2. Hermitian connection. 8.3. Riemann-Roch operator. 8.4. Weitzenb ck formula. 8.5. Index theorem. 8.6. Riemann-Roch operator in complex analysis. |