黎曼几何和几何分析（第6版） [Riemannian Geometry and Geometric Analysis Sixth Edition] epub pdf mobi txt 电子书 下载 2025

黎曼几何和几何分析（第6版） [Riemannian Geometry and Geometric Analysis Sixth Edition] epub pdf mobi txt 电子书 下载 2025

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黎曼几何和几何分析（第6版） [Riemannian Geometry and Geometric Analysis Sixth Edition] epub pdf mobi txt 电子书 下载 2025

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内容简介

　　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.

内页插图

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

前言/序言

黎曼几何和几何分析（第6版） [Riemannian Geometry and Geometric Analysis Sixth Edition] epub pdf mobi txt 电子书 下载 2025

黎曼几何和几何分析（第6版） [Riemannian Geometry and Geometric Analysis Sixth Edition] 下载 epub mobi pdf txt 电子书 2025

评分☆☆☆☆☆

经典正版图书，配送及时，非常满意

评分☆☆☆☆☆

很好的书，很好的书

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很好很适合学生老师阅读

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朋友介绍的书，挺好的

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这本书不错，是几何分析的入门教材。

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Geometry is not important for all my classes but my parents

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黎曼流形上的几何学。德国数学家G.F.B.黎曼19世纪中期提出的几何学理论。1854年黎曼在格丁根大学发表的题为《论作为几何学基础的假设》的就职演说，通常被认为是黎曼几何学的源头。在这篇演说中，黎曼将曲面本身看成一个独立的几何实体，而不是把它仅仅看作欧几里得空间中的一个几何实体。他首先发展了空间的概念，提出了几何学研究的对象应是一种多重广义量 ，空间中的点可用n个实数（x1，……，xn）作为坐标来描述。这是现代n维微分流形的原始形式，为用抽象空间描述自然现象奠定了基础。这种空间上的几何学应基于无限邻近两点（x1，x2，……xn）与（x1+dx1，……xn+dxn）之间的距离，用微分弧长度平方所确定的正定二次型理解度量。亦即 （gij）是由函数构成的正定对称矩阵。这便是黎曼度量。赋予黎曼度量的微分流形，就是黎曼流形。

评分☆☆☆☆☆

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 a

评分☆☆☆☆☆

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