Surveys in Geometry II

Preface
Contents
Editor and Contributors
	About the Editor
	Contributors
1 Introduction
2 Geometry on Surfaces, a Source for Mathematical Developments
	2.1 Introduction
	2.2 Rigidity of Geometric Structures
		2.2.1 Volume, Symplectic and Contact Forms
		2.2.2 Almost Complex Structures
		2.2.3 Almost Complex Structures on n-Spheres
	2.3 The First Compact Riemann Surface
		2.3.1 The Riemann Sphere
		2.3.2 The Group SU(2) and Its Action on the Riemann Sphere
	2.4 All Three Planar Geometries and Hyperbolic 3-Space Simultaneously
		2.4.1 A Stratification of the Riemann Sphere Arising from Algebra
		2.4.2  A Note on the Field with One Element
		2.4.3 The Riemann Sphere and Shadow Numbers
		2.4.4 J-Compatible Metrics
		2.4.5 Spherical Geometry
		2.4.6 Models for Hyperbolic 3-Space and Metrics of Constant Positive Curvature on the Sphere
		2.4.7 Models for the Hyperbolic Plane
	2.5 Uniformisation
		2.5.1 Riemann's Uniformisation of Simply Connected Domains
		2.5.2 Uniformization of Multiply Connected Domains
		2.5.3 Uniformization of Simply Connected Riemann Surfaces
	2.6 Branched Coverings
		2.6.1 The Riemann–Hurwitz Formula
		2.6.2 Speiser Curves, Nets and Line Complexes
		2.6.3 Thurston's Realization Theorem
	2.7 The Type Problem
		2.7.1 Ahlfors on the Type Problem
		2.7.2 Nevanlinna on the Type Problem
		2.7.3 Quasiconformal Mappings and Teichmüller's Work on the Type Problem
		2.7.4 Lavrentieff on the Type Problem
		2.7.5 Milnor on the Type Problem
		2.7.6 Probablistic Approaches
		2.7.7 Electricity
	2.8 Uniformization: Geometry and Combinatorics
		2.8.1 Dessins d'enfants
		2.8.2 Slalom Polynomials
		2.8.3 A Stratification of the Space of Monic Polynomials
		2.8.4 Rational Maps, Speiser Colored Cell Decompositions, Classical Knots and Links
	References
3 Teichmüller Spaces and Their Various Metrics
	3.1 Introduction
	3.2 Teichmüller Space and the Teichmüller Metric
	3.3 Teichmüller Maps and Teichmüller Discs
	3.4 Extremal Lengths and Measured Foliations
	3.5 Finsler Structure of Teichmüller Space
	3.6 Thurston's Asymmetric Metric
	3.7 The Earthquake Metric
	References
4 Double Forms, Curvature Integrals and the Gauss–Bonnet Formula
	4.1 Introduction
		4.1.1 Organization of the Chapter
	4.2 An Overview of Hopf's Problem on the Curvatura Integra
	4.3 Double Forms and Their Geometric Applications
		4.3.1 Definition and Examples
		4.3.2 Transposition and Symmetric Double Forms
		4.3.3 The Contraction Operator
		4.3.4 Integrating Double Forms
		4.3.5 The Lipschitz–Killing Curvatures
	4.4 Chern's Insight and the Tao of Gauss–Bonnet
		4.4.1 The Emergence of the Pfaffian
		4.4.2 A Tale of Exactness, Featuring a Vector Field
		4.4.3 Harvesting the Ripe Gauss–Bonnet Fruit
		4.4.4 Demystifying the Gauss–Bonnet Boundary Term Using Double Forms
	4.5 Examples and Applications
		4.5.1 Flat Manifolds
		4.5.2 Manifolds with Flat Boundary
		4.5.3 Space Forms
		4.5.4 Further Results on Non Compact Manifolds
		4.5.5 The Direct Product of a Ball and a Sphere
		4.5.6 Rotationally Symmetric Metrics
	4.6 Miscellaneous Remarks
		4.6.1 Remarks on Signs and Other Conventions
		4.6.2 Comparison with Chern's Papers
		4.6.3 Double Factorials and the Volume of Spheres
	4.7 Exercises
	Appendix A: Some Background in Differential Geometry
		A.1 Some Linear Algebra on the (co)tangent Space
		A.2 Connection and Curvature
		A.3 Moving the Frame with Élie Cartan
	References
5 Quaternions, Monge–Ampère Structures and k-Surfaces
	5.1 Introduction
		5.1.1 Introduction
	5.2 Quaternions and Bernstein-Type Theorems
		5.2.1 Quaternions
		5.2.2 Compatible Complex Structures
		5.2.3 Compatible Quaternionic Structures
		5.2.4 Calibrations
		5.2.5 The Monge–Ampère Equation
		5.2.6 Positivity I
		5.2.7 Positivity II: A Holomorphic Approach
		5.2.8 Bernstein Type Theorems I
		5.2.9 Bernstein Type Theorems II: Non-trivial Boundary
		5.2.10 Non-quadratic Solutions
	5.3 Monge–Ampère Structures and k-Surfaces
		5.3.1 Monge–Ampère Structures
		5.3.2 Compactness
		5.3.3 Applications I: k-Surfaces
		5.3.4 Applications II: Isometric Immersions
	References
6 Lagrangian Grassmannians of Polarizations
	6.1 Introduction
	6.2 Compatible Triples
	6.3 Polarizations
	6.4 The Grassmannian of Polarizations Associated with a Symplectic Form
	6.5 The Grassmannian of Polarizations Associated with an Inner Product
	6.6 Motivation for the Restricted Grassmannian
	6.7 Sewing and Diff(S1)
	6.8 Solutions
	References
7 Metric Characterizations of Projective-Metric Spaces
	7.1 Introduction
	7.2 Preliminaries
		7.2.1 Some Classes of Projective-Metric Spaces
			7.2.1.1 Classical Geometries
			7.2.1.2 Minkowski Geometries
			7.2.1.3 Hilbert Geometries
			7.2.1.4 Finslerian Projective-Metric Spaces
		7.2.2 Reference Functions and Ratios
	7.3 Metric Properties of the Space
		7.3.1 Curvature
		7.3.2 Ptolemaic Projective-Metric Spaces
		7.3.3 Symmetry of the Perpendicularity of Lines
		7.3.4 Erdős Ratio
		7.3.5 Regular Polygons
	7.4 Metrically Defined Objects
		7.4.1 Bisector of Point Pairs
		7.4.2 Median Rays of Point Pairs
		7.4.3 Equidistants of Lines
		7.4.4 Circumcenter and Orthocenter
		7.4.5 Angular Bisector
	7.5 Metric Properties of Triangles
		7.5.1 Ceva and Menelaus Property
		7.5.2 Euler's Ratio-Sum
	7.6 Metric Constructions Describing Quadratic Curves
		7.6.1 Ellipses, Circles and Hyperbolas
		7.6.2 Conics
	7.7 Discussions and Further Open Questions
	References
8 Supplement to ``Metric Characterization of Projective-Metric Spaces''
	8.1 Introduction
	8.2 Equidistants of Lines
	8.3 Ceva and Menelaus Property
	References
9 Metric Problems in Projective and Grassmann Spaces
	9.1 Introduction
	9.2 Equiangular Lines and Equi-Isoclinic Subspaces
	9.3 Introduction to Seidel Matrices
	9.4 Real Equiangular Lines
	9.5 Complex Equiangular Lines
	9.6 Superposability Order of Projective and Grassmann Spaces
	9.7 Complex Conference Matrices
		9.7.1 Real Symmetric Conference Matrices
		9.7.2 Complex Hermitian Conference Matrices
	9.8 Complex Equiangular Tight Frames
	9.9 Complex Symmetric Conference Matrices of Odd Orders
	9.10 Equi-Isoclinic Planes in Euclidean Odd Dimensional Spaces
	9.11 Equi-Isoclinic Planes in C4
		9.11.1 Triples
		9.11.2 Quadruples
		9.11.3 Quintuples
		9.11.4 Sextuples
	9.12 Quaternionic Equiangular Lines
		9.12.1 A One to One Correspondence
		9.12.2 Quaternionic Equiangular Lines in H2
	References
10 On the Geometry of Finite Homogeneous Subsets of Euclidean Spaces
	Introduction
	10.1 Homogeneous Metric Spaces and Their Special Subclasses
	10.2 Some Properties of Finite Homogeneous Metric Spaces
	10.3 Finite Homogeneous Subspaces of Euclidean Spaces
	10.4 Regular and Semiregular Polytopes
	10.5 Regular and Semiregular Polytopes with Normal Homogeneous or Clifford–Wolf homogeneous Vertex Sets
	10.6 General Results on the m-Point Homogeneity and m-Point Homogeneous Subspaces of Euclidean Spaces
	10.7 On m-Point Homogeneous Polyhedra in R3
	10.8 Some Results on the Point Homogeneity Degree
	10.9 Conclusion
	References
11 Discrete Coxeter Groups
	11.1 Introduction
	11.2 Coxeter Groups
		11.2.1 What Is a Coxeter Group?
	11.3 Hyperbolic Reflection Groups
		11.3.1 Hyperbolic Polytopes
		11.3.2 Classical Results in Dimensions 2 and 3
			11.3.2.1 Dimension 2
			11.3.2.2 Dimension 3
		11.3.3 The Gram Matrix of a Hyperbolic Polytope
		11.3.4 Lannér and Quasi-Lannér Coxeter Groups
		11.3.5 Absence in Large Dimension
		11.3.6 Hyperbolic Coxeter Polytopes with Few Facets
		11.3.7 Convex Cocompact Hyperbolic Reflection Groups
	11.4 Projective Reflection Groups
		11.4.1 Tits–Vinberg's Theorem
		11.4.2 From Cartan Matrices to Mirror Polytopes
		11.4.3 Anosov Reflection Groups
		11.4.4 Convex Cocompact Projective Reflection Groups
		11.4.5 Divisible and Quasi-Divisible Domains
		11.4.6 Cocompact Action of Coxeter Groups
	11.5 Examples of Projective Reflection Groups
		11.5.1 Kac–Vinberg's Example
		11.5.2 Benoist's Examples and More
	11.6 Hitchin Component of Polygon Groups
	11.7 Properly Discontinuous Affine Groups
		11.7.1 Auslander's Conjecture and Milnor's Question
		11.7.2 Properly Discontinuous Affine Coxeter Groups
	References
12 Isoperimetry in Finitely Generated Groups
	12.1 Introduction
	12.2 Preliminaries from Geometric Group Theory
	12.3 The Isoperimetric Inequality in Finitely Generated Groups and Some Consequences
		12.3.1 Isoperimetric Inequalities in Free Groups and Free Abelian Groups
		12.3.2 Statement of the Main Result
		12.3.3 Proof of the Main Theorem
		12.3.4 Some Consequences of the Main Result
		12.3.5 Asymptotic Estimates
		12.3.6 Final Remarks
	12.4 Some Properties and Calculations of the U-Transform
		12.4.1 Basic Properties
		12.4.2 Computing the U-Transform
		12.4.3 Asymptotic Behavior
		12.4.4 Comparison with the Legendre Transform
	References
Index