Dirac operators in Riemannian geometry / Thomas Friedrich ; translated by Andreas Nestke.
Idioma: Inglés Lenguaje original: Alemán Series Graduate studies in mathematics, v. 25Editor: Providence, R.I. : American Mathematical Society, c2000Descripción: xvi, 195 p. ; 27 cmISBN: 0821820559 (alk. paper)Títulos uniformes: Dirac-Operatoren in der Riemannschen Geometrie. English Tema(s): Geometry, Riemannian | Dirac equationOtra clasificación: 58J60 58-01 | 53C25Introduction Chapter 1. Clifford Algebras and Spin Representation [1] 1.1. Linear algebra of quadratic forms [1] 1.2. The Clifford algebra of a quadratic form [4] 1.3. Clifford algebras of real negative definite quadratic forms [10] 1.4.The pin and the spin group [14] 1.5.The spin representation [20] 1.6.The group Spinc [25] 1.7.Real and quatemionic structures in the space of n-spinors [29] 1.8.References and exercises [32] Chapter 2. Spin Structures [35] 2.1.Spin structures on SO(n)-principal bundles [35] 2.2.Spin structures in covering spaces [42] 2.3.Spin structures on G-principal bundles [45] 2.4.Existence of spinc structures [47] 2.5.Associated spinor bundles [53] 2.6.References and exercises [56] Chapter 3. Dirac Operators [57] 3.1. Connections in spinor bundles [57] 3.2. The Dirac and the Laplace operator in the spinor bundle [67] 3.3. The Schrodinger-Lichnerowicz formula [71] 3.4. Hermitian manifolds and spinors [73] 3.5. The Dirac operator of a Riemannian symmetric space [82] 3.6. References and Exercises [88] Chapter 4. Analytical Properties of Dirac Operators [91] 4.1. The essential self-adjointness of the Dirac operator in L2 [91] 4.2. The spectrum of Dirac operators over compact manifolds [98] 4.3. Dirac operators are Fredholm operators [107] 4.4. References and Exercises [111] Chapter 5. Eigenvalue Estimates for the Dirac Operator and Twistor Spinors [113] 5.1. Lower estimates for the eigenvalues of the Dirac operator [113] 5.2. Riemannian manifolds with Killing spinors [116] 5.3. The twistor equation [121] 5.4. Upper estimates for the eigenvalues of the Dirac operator [125] 5.5. References and Exercises [127] Appendix A. Seiberg-Witten Invariants [129] A.l. On the topology of 4-dimensional manifolds [129] A.2. The Seiberg-Witten equation [134] A.3. The Seiberg-Witten invariant [138] A.4. Vanishing theorems [144] A.5. The case dim [146] A.6. The Kahler case [147] A.7. References [153] Appendix B. Principal Bundles and Connections [155] B. l. Principal fibre bundles [155] 13.2. The classification of principal bundles [162] B.3. Connections in principal bundles [163] B.4. Absolute differential and curvature [166] B.5. Connections in U(1)-principal bundles and the Weyl theorem [169] B.6. Reductions of connections [173] B.7. Frobenius’ theorem [174] B.8. The Freudenthal-Yamabe theorem [177] B.9. Holonomy theory [177] B.10. References [178] Bibliography [179] Index [193]
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Introduction --
Chapter 1. Clifford Algebras and Spin Representation [1] --
1.1. Linear algebra of quadratic forms [1] --
1.2. The Clifford algebra of a quadratic form [4] --
1.3. Clifford algebras of real negative definite quadratic forms [10] --
1.4.The pin and the spin group [14] --
1.5.The spin representation [20] --
1.6.The group Spinc [25] --
1.7.Real and quatemionic structures in the space of n-spinors [29] --
1.8.References and exercises [32] --
Chapter 2. Spin Structures [35] --
2.1.Spin structures on SO(n)-principal bundles [35] --
2.2.Spin structures in covering spaces [42] --
2.3.Spin structures on G-principal bundles [45] --
2.4.Existence of spinc structures [47] --
2.5.Associated spinor bundles [53] --
2.6.References and exercises [56] --
Chapter 3. Dirac Operators [57] --
3.1. Connections in spinor bundles [57] --
3.2. The Dirac and the Laplace operator in the spinor bundle [67] --
3.3. The Schrodinger-Lichnerowicz formula [71] --
3.4. Hermitian manifolds and spinors [73] --
3.5. The Dirac operator of a Riemannian symmetric space [82] --
3.6. References and Exercises [88] --
Chapter 4. Analytical Properties of Dirac Operators [91] --
4.1. The essential self-adjointness of the Dirac operator in L2 [91] --
4.2. The spectrum of Dirac operators over compact manifolds [98] --
4.3. Dirac operators are Fredholm operators [107] --
4.4. References and Exercises [111] --
Chapter 5. Eigenvalue Estimates for the Dirac Operator and Twistor Spinors [113] --
5.1. Lower estimates for the eigenvalues of the Dirac operator [113] --
5.2. Riemannian manifolds with Killing spinors [116] --
5.3. The twistor equation [121] --
5.4. Upper estimates for the eigenvalues of the Dirac operator [125] --
5.5. References and Exercises [127] --
Appendix A. Seiberg-Witten Invariants [129] --
A.l. On the topology of 4-dimensional manifolds [129] --
A.2. The Seiberg-Witten equation [134] --
A.3. The Seiberg-Witten invariant [138] --
A.4. Vanishing theorems [144] --
A.5. The case dim [146] --
A.6. The Kahler case [147] --
A.7. References [153] --
Appendix B. Principal Bundles and Connections [155] --
B. l. Principal fibre bundles [155] --
13.2. The classification of principal bundles [162] --
B.3. Connections in principal bundles [163] --
B.4. Absolute differential and curvature [166] --
B.5. Connections in U(1)-principal bundles and the Weyl theorem [169] --
B.6. Reductions of connections [173] --
B.7. Frobenius’ theorem [174] --
B.8. The Freudenthal-Yamabe theorem [177] --
B.9. Holonomy theory [177] --
B.10. References [178] --
Bibliography [179] --
Index [193] --
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