دانلود کتاب Surveys in Geometry I

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Preface
Contents
Contributors
About the Editor
1 Introduction
2 Spherical Geometry—A Survey on Width and Thickness of Convex Bodies
2.1 Introduction
2.2 Elementary Notions
2.3 Convex Bodies
2.4 Lunes
2.5 Width and Thickness of a Convex Body
2.6 Reduced Spherical Bodies
2.7 More About Reduced Bodies on S2
2.8 Spherical Reduced Polygons
2.9 Diameter of Convex Bodies and Reduced Bodies
2.10 Bodies of Constant Width
2.11 Complete Spherical Convex Bodies
2.12 Final Remarks
References
3 Minkowski Geometry—Some Concepts and RecentDevelopments
3.1 Introduction
3.2 First Steps
3.3 Orthogonality Concepts
3.4 Some Elementary Geometry
3.5 Metric Problems
3.6 Classical and Differential Theory of Planar Curves
3.7 Differential Geometry of Surfaces
3.8 Minkowski Billiards
3.9 Other Topics
References
4 Orthogonality Types in Normed Linear Spaces
4.1 Introduction
4.2 Definitions of Orthogonalities
4.3 General Properties of Orthogonality
4.4 Roberts Orthogonality
4.5 Birkhoff Orthogonality
4.5.1 Existence and Uniqueness
4.5.2 Symmetry
4.5.3 Additivity
4.5.4 Orthogonal Diagonals
4.5.5 Birkhoff Orthogonality and Hyperplanes
4.6 Isosceles Orthogonality
4.6.1 Existence and Uniqueness
4.6.2 α-Existence and α-Uniqueness
4.6.3 Homogeneity
4.6.4 Additivity
4.6.5 Orthogonal Diagonals
4.6.6 Isosceles Orthogonality and Hyperplanes
4.7 Pythagorean Orthogonality
4.7.1 Existence and Uniqueness
4.7.2 α-Existence and α-Uniqueness
4.7.3 Homogeneity
4.7.4 Additivity
4.7.5 Orthogonal Diagonals
4.7.6 Pythagorean Orthogonality and Hyperplanes
4.8 Relations Between Birkhoff, Isosceles, and Pythagorean Orthogonality
4.8.1 Implications over the Whole Space
4.8.2 The Joly Construction
4.8.3 Implications over the Unit Sphere
4.9 Other Orthogonalities
4.9.1 Carlsson Orthogonalities
Relations of Carlsson Orthogonalities with Other Orthogonalities
4.9.2 Boussouis Orthogonalities
4.9.3 Singer Orthogonality
4.9.4 DP-Orthogonality
4.9.5 Diminnie Orthogonality
4.9.6 Area Orthogonality
4.9.7 Height Orthogonality
4.10 A Survey on Further Results
References
5 Convex Bodies: Mixed Volumes and Inequalities
5.1 Introduction
5.2 Basic Notions
5.2.1 Convex Bodies
5.2.2 Steiner Formula
5.2.3 Integral Geometry
5.3 Technical Tools
5.3.1 The Support Function
5.3.2 The Minkowski Sum and Scaling of Convex Bodies
5.3.3 The Hausdorff Distance
5.3.4 The Support Function and the Volume
5.4 Mixed Volumes and Inequalities
5.4.1 Definition and Examples
5.4.2 Alexandrov–Fenchel and Minkowski Inequalities
5.5 The Brunn–Minkowski Inequality and Symmetrization
5.5.1 The Brunn–Minkowski Inequality
5.5.2 Brunn–Minkowski Implies Minkowski
5.5.3 Steiner Symmetrization and Schwartz Rounding
5.5.4 A Couple of Other Inequalities
5.6 Isoperimetric Inequality and Laplacian Eigenvalues
5.6.1 Normal Parametrization of Convex Plane Curves
5.6.2 The Isoperimetric Inequality in the Plane
5.6.3 Higher Dimensions
5.7 A New Take on the Mixed Volumes
5.7.1 Strongly Isomorphic Polytopes and Volume Polynomials
5.7.2 Alexandrov–Fenchel Inequality Revisited
5.7.3 The Discrete Spherical Laplacian
References
6 Compactness and Finiteness Results for Gromov-HyperbolicSpaces
6.1 Introduction
6.2 Lecture 1
6.2.1 Some Assumptions
6.2.2 Geodesics
6.2.3 Gromov-Hyperbolic Spaces
6.2.4 Measures
6.2.5 Groups Acting by Isometries
6.2.6 Entropy of (X, d, μ)
6.2.7 Bishop–Gromov\'s Inequality
6.2.8 Main Theorem
6.3 Lecture 2
6.3.1 Quadrangle Inequality for δ-Hyperbolic Metric Spaces
6.3.2 A Simple Version of Theorem 6.2.8
6.3.2.1 Easy Lemma
6.3.2.2 More Technical Lemmas
6.3.2.3 Proof of Theorem 6.3.2
6.4 Lecture 3
6.4.1 Fundamental Group
6.4.2 First Homology Group
6.4.3 Bounding the First Betti Number
6.5 Lecture 4
6.5.1 Busemann Spaces
6.5.2 Packings and Coverings
6.5.3 A Contraction in Busemann Spaces
6.6 Lecture 5
6.6.1 Growth of Groups
6.6.2 A Margulis Lemma
6.6.3 The Thin-Thick Decomposition
6.7 Lecture 6
6.7.1 Systoles
6.7.2 Bounding from Below the Systole
6.8 Lecture 7
6.8.1 Marked Groups
6.8.2 Finitely Presented Groups
6.8.3 Finiteness Theorem
6.9 Lecture 8
6.9.1 CAT(0)-Spaces
6.9.2 Towards a Compactness Result
6.9.2.1 Gromov–Hausdorff Distance Between Metric Spaces
6.10 Lecture 9
6.10.1 Compactness
6.10.2 Topological Finiteness
6.10.3 Growth and Entropy of Groups
6.11 Lecture 10
6.11.1 Algebraic Entropy of Groups Acting on Hyperbolic Metric Spaces
6.11.2 Entropy of a δ-Hyperbolic Space with a Group Action
References
7 All 4-Dimensional Smooth Schoenflies Balls Are Geometrically Simply-Connected
7.1 Introduction
7.2 The Two Collapsing Flows
7.3 The New Context and the Doubling
7.4 Four Dimensional Thickenings and Compactifications
7.5 Exterior Discs
7.6 Confinement
7.7 Balancing Blue and Red
7.8 Change of Colour
References
8 Classical Differential Topology and Non-commutative Geometry
8.1 Introduction
8.2 The Construction
8.3 The Proofs
References
9 A Short Introduction to Translation Surfaces, Veech Surfaces, and Teichmüller Dynamics
9.1 Introduction
9.2 Three Different Definitions of a Translation Surface
9.2.1 The Most Hands-on Definition: Polygons with Identifications
9.2.2 Main Examples of Translation Surfaces
9.2.3 Definition Through an Atlas
9.2.4 Definition of a Translation Surface as a Holomorphic Differential
9.2.5 Definition of a Half-Translation Surface as a Quadratic Differential
9.2.6 Dynamical System Point of View
9.3 Moduli Space
9.3.1 Strata
9.3.2 Period Coordinates
9.3.3 Dimension of the Strata
9.3.4 Quadratic Differentials
9.3.4.1 Dimension of Strata of Quadratic Differentials
9.3.5 Another Look at the Dimension of Hg
9.4 The Teichmüller Geodesic Flow
9.4.1 Teichmüller\'s Theorem
9.4.2 Masur\'s Criterion
9.5 Orbits of the GL+2(R)-Action
9.6 Veech Groups
9.6.1 The Veech Group of the Square Torus is SL2 (Z )
9.6.2 Veech Groups of the Regular Polygons
9.6.3 Veech Groups of Square-Tiled Surfaces
9.7 Teichmüller Disks
9.7.1 Example: The Teichmüller Disk of the Torus
9.7.1.1 Hyperbolic Metric vs. Teichmüller Metric
9.7.1.2 Hyperbolic Geodesics
9.7.1.3 Horocycles
9.7.1.4 Relationship Between the Asymptotic Behaviour of Hyperbolic Geodesics, and the Dynamic Behaviour of Euclidean Geodesics
9.7.2 Examples of Higher Genus Teichmüller Disks
9.7.2.1 Three-Squared Surfaces
9.7.2.2 Regular Polygons
9.8 Veech Surfaces
9.8.1 The Smillie-Weiss Theorem
9.8.2 The Veech Alternative
9.8.2.1 Examples of Veech Surfaces
9.9 Classification of Orbits in Genus Two
9.9.1 Square-Tiled Surfaces
9.9.2 A Homological Detour
9.9.2.1 The Tautological Subspace
9.9.2.2 The Trace Field
9.9.2.3 The Jacobian Torus
9.9.2.4 Real Multiplication
9.10 What Is Known in Higher Genus?
References
10 Teichmüller Spaces and the Rigidity of Mapping Class Group Actions
10.1 Teichmüller Metric
10.1.1 Quasi-Conformal Homeomorphisms
10.1.2 Teichmüller Distance
10.1.3 Teichmüller Maps
10.2 Mapping Class Group Actions
10.3 Curve Complexes and Their Automorphisms
10.4 An Alternative Approach to Royden\'s Rigidity
10.5 Thurston\'s Asymmetric Metric
References
11 Holomorphic G-Structures and Foliated Cartan Geometries on Compact Complex Manifolds
11.1 Introduction
11.2 Holomorphic G-Structure
11.3 GL(2)-Geometry and SL(2)-Geometry
11.3.1 Models of the Flat Holomorphic Conformal Geometry
11.3.2 Geometry of the Quadric Q3
11.3.3 Classification Results
11.4 Holomorphic Cartan Geometry
11.4.1 Classical Case
11.4.2 Branched Cartan Geometry
11.5 Transverse Cartan Geometry
11.5.1 Foliated Atiyah Bundle Description
11.5.2 Flatness Results
11.5.2.1 Rationally Connected Manifolds
11.5.2.2 Simply Connected Calabi–Yau Manifolds
11.5.3 A Topological Criterion
11.6 Some Related Open Problems
References
Index