黎曼幾何和幾何分析

內(nèi)容概要

本書是一部值得一讀的研究生教材，內(nèi)容主要涉及黎曼幾何基本定理的研究，如霍奇定理、Rauch比較定理、Lyusternik和Fet定理調(diào)和映射的存在性等，書中還有當(dāng)代數(shù)學(xué)研究領(lǐng)域中的最熱門論題，有些內(nèi)容則是首次出現(xiàn)在教科書中。本書各章有習(xí)題。    目次: 基本理論; 德拉姆上同調(diào)和調(diào)和微分形式；并行傳輸、聯(lián)絡(luò)和共變導(dǎo)數(shù)；測(cè)地學(xué)和雅可比場(chǎng)；對(duì)稱空間和Kahler流形；莫斯理論和Floer同調(diào)；量子場(chǎng)論中的變分問題；調(diào)和映射。    讀者對(duì)象：數(shù)學(xué)和理論物理專業(yè)的研究生、教師和科研人員。

書籍目錄

1.Foundational Material  1.1 Manifolds and Differentiable Manifolds  1.2 Tangent Spaces  1.3 Submanifolds  1.4 Riemannian Metrics  1.5 Vector Bundles  1.6 Integral Curves of Vector Fields.Lie Algebras  1.7 Lie Groups  1.8 Spin Structures  Exercises for Chapter 12.De Rham Cohomology and Harmonic Differential Forms  2.1 The Laplace Operator  2.2 Representing Cohomology Classes by Harmonic Forms  2.3 Generalizations  Exercises for Chapter 23.Parallel Transport,Connections,and Covariant Derivatives  3.1 Connections in Vector Bundles  3.2 Metric Connections.The Yang-Mills Functional  3.3 The Levi-Civita Connection  3.4 Coonections for Spin Structures and the Dirac Operator  3.5 The Bochner Method  3.6 The Geometry of Submanifolds.Minimal Submanifolds  Exercises for Chapter 34.Geodesics and Jacobi Fields  4.1 1st and 2nd Variation of Arc Length and Energy  4.2 Jacobi Fields  4.3 Conjugate Points and Distance Minimizing Geodesics  4.4 Riemannian Manifolds of Constant Curvature  4.5 The Rauch Comparison Theorems and Other Jacobi Field Estimates  4.6 Geometric Applications of Jacobi Field Estimates  4.7 Approximate Fundamental Solutions and Representation Formulae  4.8 The Geometry of Manifolds of Nonpositive Sectional Curvature  Exercises for Chapter 4A Short Survey on Curvature and Topology5.Symmetric Spaces and Kahler Manifolds6.Morse Theory and Floer Homology7.Variational Problems from Quantum Field Theory8.Harmonic MapsAppendix A:Linear Elliptic Partial Differential EquationAppendix B:Fundamental Groups and Covering SpacesIndex

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