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E-raamat: Index Theorem And The Heat Equation Method, The

(Suzhou Univ, China), Series edited by (Chern Inst Of Mathematics, China)
  • Formaat: 308 pages
  • Sari: Nankai Tracts in Mathematics 2
  • Ilmumisaeg: 02-Jul-2001
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • Keel: eng
  • ISBN-13: 9789814491112
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Index Theorem And The Heat Equation Method, TheSuurem pilt
  • Formaat: 308 pages
  • Sari: Nankai Tracts in Mathematics 2
  • Ilmumisaeg: 02-Jul-2001
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • Keel: eng
  • ISBN-13: 9789814491112
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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.
Preface v
Definitions and Formulas ix
Preliminaries in Riemannian Geometry
1(82)
Basic Notions of Riemannian Geometry
1(10)
Computations by using Orthonormal Moving Frame
11(3)
Differential Forms and Orthonormal Moving Frame Method
14(9)
Classical Geometric Operators
23(18)
Normal Coordinates
41(19)
Computations on Sphere
60(10)
Connections on Vector Bundles and Principal Bundles
70(8)
General Tensor Calculus
78(5)
Schrodinger and Heat Operators
83(32)
Fundamental Solution and Levi Iteration
84(5)
Existence of Fundamental Solution
89(5)
Cauchy Problem of Heat Equation
94(4)
Hodge Theorem
98(8)
Applications of Hodge Theorem
106(6)
Index Problem
112(3)
MP Parametrix and Applications
115(24)
MP Parametrix
115(4)
Existence of Initial Solutions
119(8)
Asymptotic Expansion for Heat Kernel
127(3)
Local Index for Elliptic Operators
130(9)
Chern-Weil Theory
139(36)
Characteristic Forms and Characteristic Classes
139(11)
General Characteristic Forms
150(12)
Chern Root Algorithm
162(3)
Formal Approach to Local Index of Signature Operator
165(10)
Clifford Algebra and Super Algebra
175(20)
Clifford Algebra
175(8)
Super Algebra
183(7)
Computations on Supertraces
190(5)
Dirac Operator
195(18)
Spin Structure
195(6)
Spinor Bundle
201(2)
Dirac Operator
203(4)
Index of Dirac Operator
207(6)
Local Index Theorems
213(32)
Local Index Theorem for Dirac Operator
214(17)
Local Index Theorem for Signature Operator
231(10)
Local Index Theorem for de Rham-Hodge Operator
241(4)
Riemann-Roch Theorem
245(34)
Hermitian Metric
245(4)
Hermitian Connection
249(10)
Riemann-Roch Operator
259(9)
Weitzenbock Formula
268(5)
Index Theorem
273(3)
Riemann-Roch Operator in Complex Analysis
276(3)
References 279(4)
Index 283
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