作者：[美]彼得.B.吉尔基
页数：497
出版社：哈尔滨工业大学出版社
出版日期：2022
ISBN：9787576704051
电子书格式：pdf/epub/txt
内容简介
本书是一部探讨分析学、几何学与拓扑之间关系的英文版学术专著。这本书探讨了分析学、几何学与拓扑之间的关系.除非另有说明，否则我们将假设本书中讨论的任何流形都是紧的且没有边界的，即它是一个闭流形。我们通常(但不总是)假设闭流形是连通的。我们的工作在光滑(比如C∞)类中进行.η不变量是K理论和等变(下）配边的R/Z值谱不变量，它也在研究正标量曲率度量中发挥主要的作用.我们将关注球面空间形式和球面空间形式群；分析学与拓扑之间的相互作用是由几何学调节的.球面空间形式是具有恒定正截面曲率的流形；球面空间形式群是同调维数为1的群.因此，这些流形和群本身就具有内在的意义。
目录
Preface
1. Partial differential operators
1.1 Characteristic classes
1.1.1 Cohomology groups
1.1.2 Vector bundles
1.1.3 Chern-Weil theory
1.1.4 The Chern classes
1.1.5 The Todd, A, and Hirzebruch L polynomials
1.1.6 The Stiefel-Whitney classes
1.1.7 K theory and the Chern character
1.2 Clifford algebras and spinors
1.2.1 Clifford algebras
1.2.2 Principal G bundles
1.2.3 Spin structures on vector bundles
1.2.4 The spin bundles
1.2.5 The spin connection
1.2.6 The Dirac operator and the Lichnerowicz formula
1.3 Spectral theory of self-adjoint elliptic operators
1.3.1 Basic notational conventions
1.3.2 The Fourier transform and Sobolev spaces
1.3.3 Partial differential operators
1.3.4 Pseudo-differential operators on Euclidean space
1.3.5 Pseudo-differential operators on compact manifolds
1.3.6 Elliptic operators
1.3.7 The parametrix
1.3.8 Spectral theory
1.3.9 The Hodge decomposition theorem
1.3.10 The Hodge-de Rham theorem
1.4 The heat equation
1.4.1 Smooth kernels
1.4.2 Asymptotics of the heat equation
1.4.3 A local formula for the index of an elliptic operator
1.5 The Atiyah-Singer Index Theorem
1.5.1 The de Rham complex
1.5.2 The Hirzebrueh Signature Theorem
1.5.3 The spin complex
1.5.4 The spin-e complex
1.5.5 The Dolbeault complex
1.5.6 Tile geometrical index theorem
1.5.7 Product formulas
1.5.8 The general Atiyah-Singer index theorem
1.5.9 The Bott bundle
1.6 The eta invariant and zeta functions
1.6.1 Local invariants
1.6.2 The Mellin transform
1.6.3 The poles of the eta function
1.6.4 The normalized eta residue
1.7 Regularity of the eta function at the origin
1.7.1 A result in invariance theory
1.7.2 Analytical facts concerning the eta residue
1. Partial differential operators
1.1 Characteristic classes
1.1.1 Cohomology groups
1.1.2 Vector bundles
1.1.3 Chern-Weil theory
1.1.4 The Chern classes
1.1.5 The Todd, A, and Hirzebruch L polynomials
1.1.6 The Stiefel-Whitney classes
1.1.7 K theory and the Chern character
1.2 Clifford algebras and spinors
1.2.1 Clifford algebras
1.2.2 Principal G bundles
1.2.3 Spin structures on vector bundles
1.2.4 The spin bundles
1.2.5 The spin connection
1.2.6 The Dirac operator and the Lichnerowicz formula
1.3 Spectral theory of self-adjoint elliptic operators
1.3.1 Basic notational conventions
1.3.2 The Fourier transform and Sobolev spaces
1.3.3 Partial differential operators
1.3.4 Pseudo-differential operators on Euclidean space
1.3.5 Pseudo-differential operators on compact manifolds
1.3.6 Elliptic operators
1.3.7 The parametrix
1.3.8 Spectral theory
1.3.9 The Hodge decomposition theorem
1.3.10 The Hodge-de Rham theorem
1.4 The heat equation
1.4.1 Smooth kernels
1.4.2 Asymptotics of the heat equation
1.4.3 A local formula for the index of an elliptic operator
1.5 The Atiyah-Singer Index Theorem
1.5.1 The de Rham complex
1.5.2 The Hirzebrueh Signature Theorem
1.5.3 The spin complex
1.5.4 The spin-e complex
1.5.5 The Dolbeault complex
1.5.6 Tile geometrical index theorem
1.5.7 Product formulas
1.5.8 The general Atiyah-Singer index theorem
1.5.9 The Bott bundle
1.6 The eta invariant and zeta functions
1.6.1 Local invariants
1.6.2 The Mellin transform
1.6.3 The poles of the eta function
1.6.4 The normalized eta residue
1.7 Regularity of the eta function at the origin
1.7.1 A result in invariance theory
1.7.2 Analytical facts concerning the eta residue
