《黎曼幾何和幾何分析》是2008年3月1日世界圖書出版公司出版的圖書，作者是（德國）約斯特，本書內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定理調和映射的存在性等，書中還有當代數學研究領域中的最熱門論題，有些內容則是首次出現在教科書中。
基本介紹
- 書名：黎曼幾何和幾何分析
- 作者：(德國)約斯特
- ISBN：9787506291927
- 頁數： 566頁
- 出版社：世界圖書出版公司
- 出版時間：2008年3月1日
- 裝幀：平裝
- 開本： 24
內容簡介,目錄,
內容簡介
《黎曼幾何和幾何分析(第4版)》是一部值得一讀的研究生教材（全英文版）《黎曼幾何和幾何分析(第4版)》各章均附有習題。
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 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
