度量几何学教程(英文版)(精)/美国数学会经典影印系列

“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。 德米特里·布拉戈、尤里·布拉戈著的《度量几 何学教程(英文版)(精)》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-Caratheodory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。书中还包括大量习题，这些习题是本 书至关重要的一部分。

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