The Index Theorem and The Heat Equation Method.

This volume provides a self-contained represenation 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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Detalles Bibliográficos
Clasificación:Libro Electrónico
Formato: Electrónico eBook
Idioma:Inglés
Publicado: World Scientific 2001.
Temas:
Atiyah-Singer index theorem.
Heat equation.
Théorème d'Atiyah-Singer.
Équation de la chaleur.
Atiyah-Singer index theorem
Heat equation
Acceso en línea:Texto completo
Tabla de Contenidos:
  • PREFACE; CONTENTS; DEFINITIONS AND FORMULAS; CHAPTER 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; CHAPTER 2 SCHRODINGER 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 Theorem2.5 Applications of Hodge Theorem; 2.6 Index Problem; CHAPTER 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 Operators; CHAPTER 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; CHAPTER 5 CLIFFORD ALGEBRA AND SUPER ALGEBRA; 5.1 Clifford Algebra; 5.2 Super Algebra; 5.3 Computations on Supertraces; CHAPTER 6 DIRAC OPERATOR
  • 6.1 Spin Structure6.2 Spinor Bundle; 6.3 Dirac Operator; 6.4 Index of Dirac Operator; CHAPTER 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; CHAPTER 8 RIEMANN-ROCH THEOREM; 8.1 Hermitian Metric; 8.2 Hermitian Connection; 8.3 Riemann-Roch Operator; 8.4 Weitzenbock Formula; 8.5 Index Theorem; 8.6 Riemann-Roch Operator in Complex Analysis; REFERENCES; INDEX

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