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
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