黎曼几何与几何分析 Riemannian Geometry and Geometric Analysis Jurgen Jost 4th[PDF]

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资源信息： 中文名: 黎曼几何与几何分析 原名: Riemannian Geometry and Geometric Analysis 别名: Universitext 作者: Jurgen Jost 资源格式: PDF 版本: 4th 出版社: Springer 书号: 3-540-42627-2 发行时间: 2000年 地区: 大陆,美国,英国,法国,德国 语言: 英文 概述:

这本书我在我的另一个资源里发过，现在单独拿出来发一次。初学黎曼几何的读者可以去这里看看：http://www.VeryCD.com/topics/2817268/ 这部书是Springer出版社Universitext系列的一本，过后我会整理UT的书发到VC上来。 这是一本关于黎曼几何，尤其是几何分析的优秀教材，但是书中关于纯黎曼几何的讲述很简洁，不太适合初学者，书中很大篇幅在讲几何分析的基本方法，适合对黎曼几何已经有一定了解想从分析方面深入研究的读者。 这是书中部分前言： Riemannian geometry is characterized, and research is oriented towards and shaped by concepts (geodesics, connections, curvature, ...) and objectives, in particular to understand certain classes of (compact) Riemannian manifolds defined by curvature conditions (constant or positive or negative curvature, ...). By way of contrast, geometric analysis is a perhaps somewhat less systematic collection of techniques, for solving extremal problems naturally arising in geometry and for investigating and characterizing their solutions. It turns out that the two fields complement each other very well; geometric analysis offers tools for solving difficult problems in geometry, and Riemannian geometry stimulates progress in geometric analysis by setting ambitious goals. 　　It is the aim of this book to be a systematic and comprehensive introduction to Riemannian geometry and a representative introduction to the methods of geometric analysis. It attempts a synthesis of geometric and analytic methods in the study of Riemannian manifolds.

目录: 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 1 2. De Rham Cohomology and Harmonic Differential Forms 2.1 The Laplace Operator 2.2 Representing Co homology Classes by Harmonic Forms 2.3 Generalizations Exercises for Chapter 2 3. 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 Connections for Spin Structures and the Dirac Operator .. 3.5 The Bochner Method 3.6 The Geometry of Submanifolds. Minimal Submanifolds ... Exercises for Chapter 3 4. 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 4 A Short Survey on Curvature and Topology 5. Symmetric Spaces and Kahler Manifolds 5.1 Complex Projective Space 5.2 Kahler Manifolds 5.3 The Geometry of Symmetric Spaces 5.4 Some Results about the Structure of Symmetric Spaces .. 5.5 The Space SI(n,R)/SO(n,R) 5.6 Symmetric Spaces of Noncompact Type as Examples of Nonpositively Curved Riemannian Manifolds Exercises for Chapter 5 6. Morse Theory and Floer Homology 6.1 Preliminaries: Aims of Morse Theory 6.2 Compactness: The Palais-Smale Condition and the Existence of Saddle Points 6.3 Local Analysis: Nondegeneracy of Critical Points, Morse Lemma, Stable and Unstable Manifolds 6.4 Limits of Trajectories of the Gradient Flow 6.5 The Morse-Smale-Floer Condition: Transversality and Z2-Cohomology 6.6 Orientations and Z-homology 6.7 Homotopies 6.8 Graph flows 6.9 Orientations 6.10 The Morse Inequalities 6.11 The Palais-Smale Condition and the Existence of Closed Geodesics Exercises for Chapter 6 7. Variational Problems from Quantum Field Theory .. 7.1 The Ginzburg-Landau Functional 7.2 The Seiberg-Witten Functional Exercises for Chapter 7 8. Harmonic Maps Appendix Bibliography Index