度量幾何學教程(英文版) [A Course in Metric Geometry] pdf epub mobi txt 電子書 下載 2024
內容簡介
“度量幾何”是建立在拓撲空間長度概念基礎之上的處理幾何的方法,這種方法在*近幾十年飛速發展,並滲透到諸如群論、動力係統和偏微分方程等其他數學學科。
《度量幾何學教程(英文版)》有兩個目標:詳細闡述長度空間理論中使用的基本概念和技巧,以及為大量不同的幾何論題提供一個初等導引,這些論題都與距離觀念相關,包括黎曼度量和Carnot-Caratheodory度量、雙麯平麵、距離-體積不等式、(大規模的、粗糙的)漸近幾何、Gromov雙麯空間、度量空間的收斂性以及Alexandrov空間(非正和非負的彎麯空間)。作者傾嚮於用“易於看見”的方法來處理“易於觸碰”的數學對象。
作者設定瞭一個具有挑戰性的目標,即讓《度量幾何學教程(英文版)》的核心部分能為一年級研究生所接受。大多數新的概念和方法都按*簡單的情形來提齣並闡明,從而避免瞭技術性的障礙。書中還包括大量習題,這些習題是《度量幾何學教程(英文版)》至關重要的一部分。
內頁插圖
目錄
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
度量幾何學教程(英文版) [A Course in Metric Geometry] 下載 mobi epub pdf txt 電子書