度量几何学教程（影印版）
￥169.00定价
作者： Dmitri Burago等
出版时间：2017-01
出版社：高等教育出版社
- 高等教育出版社
- 9787040469080
- 1版
- 80375
- 46245714-4
- 精装
- 16开
- 2017-01
- 610
- 415
- 理学
- 数学
- O18
- 数学类
- 研究生（硕士、EMBA、MBA、MPA、博士）
作者简介
内容简介
“度量几何”是建立在拓扑空间长度概念基础之上的处理几何的方法，这种方法在最近几十年飞速发展，并渗透到诸如群论、动力系统和偏微分方程等其他数学学科。
这本研究生教材有两个目标：详细阐述长度空间理论中使用的基本概念和技巧，以及更一般地，为大量不同的几何论题提供一个初等导引，这些论题都与距离观念相关，包括黎曼度量和 Carnot-Carathéodory 度量、双曲平面、距离-体积不等式、(大规模的、粗糙的) 渐近几何、Gromov 双曲空间、度量空间的收敛性，以及 Alexandrov 空间 (非正和非负的弯曲空间)。作者倾向于用“易于看见”的方法来处理“易于触碰”的数学对象。
作者设定了一个具有挑战性的目标，即让本书的核心部分能为一年级研究生所接受。大多数新的概念和方法都按最简单的情形来提出并阐明，从而避免了技术性的障碍。本书还包括大量习题，这些习题是本书至关重要的一部分。
目录
Preface
Chapter 1． Metric Spaces
1．1． Definitions
1．2． Examples
1．3． Metrics and Topology
1．4． Lipschitz Maps
1．5． Complete Spaces
1．6． Compact Spaces
1．7． Hausdorff Measure and Dimension
Chapter 2． Length Spaces
2．1． Length Structures
2．2． First Examples of Length Structures
2．3． Length Structures Induced by Metrics
2．4． Characterization of Intrinsic Metrics
2．5． Shortest Paths
2．6． Length and Hausdorff Measure
2．7． Length and Lipschitz Speed
Chapter 3． Constructions
3．1． Locality， Gluing and Maximal Metrics
3．2． Polyhedral Spaces
3．3． Isometries and Quotients
3．4． Local Isometries and Coverings
3．5． Arcwise Isometries
3．6． Products and Cones
Chapter 4． Spaces of Bounded Curvature
4．1． Definitions
4．2． Examples
4．3． Angles in Alexandrov Spaces and Equivalence of Definitions
4．4． Analysis of Distance Functions
4．5． The First Variation Formula
4．6． Nonzero Curvature Bounds and Globalization
4．7． Curvature of Cones
Chapter 5． Smooth Length Structures
5．1． Riemannian Length Structures
5．2． Exponential Map
5．3． Hyperbolic Plane
5．4． Sub-Riemannian Metric Structures
5．5． Riemannian and Finsler Volumes
5．6． Besikovitch Inequality
Chapter 6． Curvature of Riemannian Metrics
6．1． Motivation： Coordinate Computations
6．2． Covariant Derivative
6．3． Geodesic and Gaussian Curvatures
6．4． Geometric Meaning of Gaussian Curvature
6．5． Comparison Theorems
Chapter 7． Space of Metric Spaces
7．1． Examples
7．2． Lipschitz Distance
7．3． Gromov-Hausdorff Distance
7．4． Gromov-Hausdorff Convergence
7．5． Convergence of Length Spaces
Chapter 8． Large-scale Geometry
8．1． Noncompact Gromov-Hausdorff Limits
8．2． Tangent and Asymptotic Cones
8．3． Quasi-isometries
8．4． Gromov Hyperbolic Spaces
8．5． Periodic Metrics
Chapter 9． Spaces of Curvature Bounded Above
9．1． Definitions and Local Properties
9．2． Hadamard Spaces
9．3． Fundamental Group of a Nonpositively Curved Space
9．4． Example： Semi-dispersing Billiards
Chapter 10． Spaces of Curvature Bounded Below
10．1． One More Definition
10．2． Constructions and Examples
10．3． Toponogov's Theorem
10．4． Curvature and Diameter
10．5． Splitting Theorem
10．6． Dimension and Volume
10．7． Gromov-Hausdorff Limits
10．8． Local Properties
10．9． Spaces of Directions and Tangent Cones
10．10． Further Information
Bibliography
Index
Chapter 1． Metric Spaces
1．1． Definitions
1．2． Examples
1．3． Metrics and Topology
1．4． Lipschitz Maps
1．5． Complete Spaces
1．6． Compact Spaces
1．7． Hausdorff Measure and Dimension
Chapter 2． Length Spaces
2．1． Length Structures
2．2． First Examples of Length Structures
2．3． Length Structures Induced by Metrics
2．4． Characterization of Intrinsic Metrics
2．5． Shortest Paths
2．6． Length and Hausdorff Measure
2．7． Length and Lipschitz Speed
Chapter 3． Constructions
3．1． Locality， Gluing and Maximal Metrics
3．2． Polyhedral Spaces
3．3． Isometries and Quotients
3．4． Local Isometries and Coverings
3．5． Arcwise Isometries
3．6． Products and Cones
Chapter 4． Spaces of Bounded Curvature
4．1． Definitions
4．2． Examples
4．3． Angles in Alexandrov Spaces and Equivalence of Definitions
4．4． Analysis of Distance Functions
4．5． The First Variation Formula
4．6． Nonzero Curvature Bounds and Globalization
4．7． Curvature of Cones
Chapter 5． Smooth Length Structures
5．1． Riemannian Length Structures
5．2． Exponential Map
5．3． Hyperbolic Plane
5．4． Sub-Riemannian Metric Structures
5．5． Riemannian and Finsler Volumes
5．6． Besikovitch Inequality
Chapter 6． Curvature of Riemannian Metrics
6．1． Motivation： Coordinate Computations
6．2． Covariant Derivative
6．3． Geodesic and Gaussian Curvatures
6．4． Geometric Meaning of Gaussian Curvature
6．5． Comparison Theorems
Chapter 7． Space of Metric Spaces
7．1． Examples
7．2． Lipschitz Distance
7．3． Gromov-Hausdorff Distance
7．4． Gromov-Hausdorff Convergence
7．5． Convergence of Length Spaces
Chapter 8． Large-scale Geometry
8．1． Noncompact Gromov-Hausdorff Limits
8．2． Tangent and Asymptotic Cones
8．3． Quasi-isometries
8．4． Gromov Hyperbolic Spaces
8．5． Periodic Metrics
Chapter 9． Spaces of Curvature Bounded Above
9．1． Definitions and Local Properties
9．2． Hadamard Spaces
9．3． Fundamental Group of a Nonpositively Curved Space
9．4． Example： Semi-dispersing Billiards
Chapter 10． Spaces of Curvature Bounded Below
10．1． One More Definition
10．2． Constructions and Examples
10．3． Toponogov's Theorem
10．4． Curvature and Diameter
10．5． Splitting Theorem
10．6． Dimension and Volume
10．7． Gromov-Hausdorff Limits
10．8． Local Properties
10．9． Spaces of Directions and Tangent Cones
10．10． Further Information
Bibliography
Index
