黎曼幾何和幾何分析（第6版） [Riemannian Geometry and Geometric Analysis Sixth Edition] epub pdf mobi txt 電子書 下載 2024

黎曼幾何和幾何分析（第6版） [Riemannian Geometry and Geometric Analysis Sixth Edition] epub pdf mobi txt 電子書 下載 2024

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黎曼幾何和幾何分析（第6版） [Riemannian Geometry and Geometric Analysis Sixth Edition] epub pdf mobi txt 電子書 下載 2024

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內容簡介

　　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.

內頁插圖

目錄

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

黎曼幾何和幾何分析（第6版） [Riemannian Geometry and Geometric Analysis Sixth Edition] epub pdf mobi txt 電子書 下載 2024

黎曼幾何和幾何分析（第6版） [Riemannian Geometry and Geometric Analysis Sixth Edition] 下載 epub mobi pdf txt 電子書

評分☆☆☆☆☆

黎曼認識到度量隻是加到流形上的一種結構，並且在同一流形上可以有許多不同的度量。黎曼以前的數學傢僅知道三維歐幾裏得空間E3中的麯麵S上存在誘導度量ds2=Edu2+2Fdudv+Gdv2，即第一基本形式，而並未認識到S還可以有獨立於三維歐幾裏得幾何賦予的度量結構。黎曼意識到區分誘導度量和獨立的黎曼度量的重要性，從而擺脫瞭經典微分幾何麯麵論中局限於誘導度量的束縛，創立瞭黎曼幾何學，為近代數學和物理學的發展作齣瞭傑齣貢獻。

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