黎曼几何和几何分析 : 第6版
暂无封面,等待上传

黎曼几何和几何分析 : 第6版

(德) 约斯特 (Jost,J.) , 著

出版社:世界图书出版公司北京公司

年代:2014

定价:99.0

书籍简介:

本书是一部值得一读的研究生教材,内容主要涉及黎曼几何基本定理的研究,如霍奇定理、Rauch比较定理、Lyusternik和Fet定理调和映射的存在性等,书中还有当代数学研究领域中的最热门论题,有些内容则是首次出现在教科书中。本书各章均附有习题。目次: 基本理论; 德拉姆上同调和调和微分形式;并行传输、联络和共变导数;测地学和雅可比场;对称空间和Kahler流形;莫斯理论和Floer同调;量子场论中的变分问题;调和映射。读者对象:数学和理论物理专业的研究生、教师和科研人员。

书籍目录:

1 Riemannian Manifolds

1.1 Manifolds and Differentiable Manifolds

1.2 Tangent Spaces

1.3 Submanifolds

1.4 Riemannian Metrics

1.5 Existence of Geodesics on Compact Manifolds

1.6 The Heat Flow and the Existence of Geodesics

1.7 Existence of Geodesics on Complete Manifolds

Exercises for Chapter 1

2 Lie Groups and Vector Bundles

2.1 Vector Bundles

2.2 Integral Curves of Vector Fields.Lie Algebras

2.3 Lie Groups

2.4 Spin Structures

Exercises for Chapter 2

3 The Laplace Operator and Harmonic Differential Forms

3.1 The Laplace Operator on Functions

3.2 The Spectrum of the Laplace Operator

3.3 The Laplace Operator on Forms

3.4 Representing Cohomology Classes by Harmonic Forms

3.5 Generalizations

3.6 The Heat Flow and Harmonic Forms

Exercises for Chapter 3

4 Connections and Curvature

4.1 Connections in Vector Bundles

4.2 Metric Connections.The Yang—Mills Functional

4.3 The Levi—Civita Connection

4.4 Connections for Spin Structures and the Dirac Operator

4.5 The Bochner Method

4.6 Eigenvalue Estimates by the Method of Li—Yau

4.7 The Geometry of Submanifolds

4.8 Minimal Submanifolds

Exercises for Chapter 4

5 Geodesics and Jacobi Fields

5.1 First and second Variation of Arc Length and Energy

5.2 Jacobi Fields

5.3 Conjugate Points and Distance Minimizing Geodesics

5.4 Riemannian Manifolds of Constant Curvature

5.5 The Rauch Comparison Theorems and Other Jacobi Field Estimates

5.6 Geometric Applications of Jacobi Field Estimates

5.7 Approximate Fundamental Solutions and Representation Formulas

5.8 The Geometry of Manifolds of Nonpositive Sectional Curvature

Exercises for Chapter 5

A Short Survey on Curvature and Topology

6 Symmetric Spaces and Kahler Manifolds

6.1 Complex Projective Space

6.2 Kahler Manifolds

6.3 The Geometry of Symmetric Spaces

6.4 Some Results about the Structure of Symmetric Spaces

6.5 The Space Sl(n,IR)/SO(n,IR)

6.6 Symmetric Spaces of Noncompact Type

Exercises for Chapter 6

7 Morse Theory and Floer Homology

7.1 Preliminaries: Aims of Morse Theory

7.2 The Palais—Smale Condition,Existence of Saddle Points

7.3 Local Analysis

7.4 Limits of Trajectories of the Gradient Flow

7.5 Floer Condition,Transversality and Z2—Cohomology

7.6 Orientations and Z—homology

7.7 Homotopies

7.8 Graph flows

7.9 Orientations

7.10 The Morse Inequalities

7.11 The Palais—Smale Condition and the Existence of Closed Geodesics

Exercises for Chapter 7

8 Harmonic Maps between Riemannian Manifolds

8.1 Definitions

8.2 Formulas for Harmonic Maps.The Bochner Technique

8.3 The Energy Integral and Weakly Harmonic Maps

8.4 Higher Regularity

8.5 Existence of Harmonic Maps for Nonpositive Curvature

8.6 Regularity of Harmonic Maps for Nonpositive Curvature

8.7 Harmonic Map Uniqueness and Applications

Exercises for Chapter 8

9 Harmonic Maps from Riemann Surfaces

9.1 Two—dimensional Harmonic Mappings

9.2 The Existence of Harmonic Maps in Two Dimensions

9.3 Regularity Results

Exercises for Chapter 9

10 Variational Problems from Quantum Field Theory

10.1 The Ginzburg—Landau Functional

10.2 The Seiberg—Witten Functional

10.3 Dirac—harmonic Maps

Exercises for Chapter 10

A Linear Elliptic Partial Differential Equations

A.1 Sobolev Spaces

A.2 Linear Elliptic Equations

A.3 Linear Parabolic Equations

B Fundamental Groups and Covering Spaces

Bibliography

Index

内容摘要:

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.
  The present work is the sixth edition of my textbook on Riemannian geometry and geometric analysis. It has developed on the basis of several graduate courses I taught at the Ruhr~University Bochum and the University of Leipzig. The main new feature of the present edition is a systematic presentation of the spectrum of the Laplace operator and its relation with the geometry of the underlying Riemannian marufold. Naturally, I have also included several smaller additions and minor corrections (for which I am grateful to several readers). Moreover, the organization of the chapters has been systematically rearranged.

书籍规格:

书籍详细信息
书名黎曼几何和几何分析 : 第6版站内查询相似图书
9787510084447
如需购买下载《黎曼几何和几何分析 : 第6版》pdf扫描版电子书或查询更多相关信息,请直接复制isbn,搜索即可全网搜索该ISBN
出版地北京出版单位世界图书出版公司北京公司
版次影印本印次1
定价(元)99.0语种英文
尺寸23 × 15装帧平装
页数印数

书籍信息归属:

黎曼几何和几何分析 : 第6版是世界图书出版公司北京公司于2014.8出版的中图分类号为 O18 ,O186.12 的主题关于 黎曼几何-研究生-教材-英文 ,几何-数学分析-研究生-教材-英文 的书籍。