Foundations of Hyperbolic Manifolds 3rd Edition by John Ratcliffe ISBN 9783030315979 3030315975

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ISBN 10: 3030315975
ISBN 13: 9783030315979
Author: John Ratcliffe

Foundations of Hyperbolic Manifolds 3rd Edition Table of contents:

CHAPTER 1: Euclidean Geometry

§1.1 Euclid’s Parallel Postulate

Spherical-Hyperbolic Duality

Curvature

§1.2 Independence of the Parallel Postulate

The Conformal Disk Model

The Upper Half-Plane Model

The Hyperboloid Model

§1.3 Euclidean n-space

Metric Spaces

Isometries

Orthogonal Transformations

Group Actions

Characterization of Euclidean Isometries

Similarities

§1.4 Geodesics

Geodesic Lines

§1.5 Arc Length

Euclidean Arc Length

§1.6 Historical Notes

CHAPTER 2: Spherical Geometry

§2.1 Spherical n-space

Cross Products

The Spherical Metric

Spherical Geodesics

§2.2 Elliptic n-space

Gnomonic Projection

§2.3 Spherical Arc Length

§2.4 Spherical Volume

§2.5 Spherical Trigonometry

Area of Spherical Triangles

§2.6 Historical Notes

CHAPTER 3: Hyperbolic Geometry

§3.1 Lorentzian n-space

Lorentz Transformations

The Time-Like Angle between Time-Like Vectors

§3.2 Hyperbolic n-space

Lorentzian Cross Products

Hyperbolic Geodesics

Hyperplanes

The Space-Like Angle between Space-Like Vectors

The Time-Like Angle between Space-Like Vectors

The Angle between Space-Like and Time-Like Vectors

§3.3 Hyperbolic Arc Length

§3.4 Hyperbolic Volume

§3.5 Hyperbolic Trigonometry

Area of Hyperbolic Triangles

Triangles with One Ideal or Ultra-Ideal Vertex

Almost Rectangular Quadrilaterals and Pentagons

Right-Angled Hyperbolic Hexagons

§3.6 Historical Notes

CHAPTER 4: Inversive Geometry

§4.1 Reflections

Inversions

Conformal Transformations

§4.2 Stereographic Projection

Cross Ratio

§4.3 Möbius Transformations

The Isometric Sphere

Preservation of Spheres

§4.4 Poincaré Extension

Möbius Transformations of Upper Half-Space

Möbius Transformations of the Unit n-Ball

§4.5 The Conformal Ball Model

Hyperbolic Translation

§4.6 The Upper Half-Space Model

§4.7 Classification of Transformations

Elliptic Transformations

Parabolic Transformations

Hyperbolic Transformations

§4.8 Historical Notes

CHAPTER 5: Isometries of Hyperbolic Space

§5.1 Topological Groups

The General Linear Group

The Unitary Group

Quotient Topological Groups

§5.2 Groups of Isometries

Groups of Möbius Transformations

§5.3 Discrete Groups

Discontinuous Groups

§5.4 Discrete Euclidean Groups

Commutativity in Discrete Euclidean Groups

§5.5 Elementary Groups

Elementary Groups of Elliptic Type

Elementary Groups of Parabolic Type

Elementary Groups of Hyperbolic Type

Solvable Groups

§5.6 Historical Notes

CHAPTER 6: Geometry of Discrete Groups

§6.1 The Projective Disk Model

§6.2 Convex Sets

Sides of a Convex Set

§6.3 Convex Polyhedra

Faces of a Convex Polyhedron

Vertices of a Convex Polyhedron

§6.4 Geometry of Convex Polyhedra

Dihedral Angles

Links of a Convex Polyhedron

§6.5 Polytopes

Generalized Polytopes

Regular Polytopes

Schläi Symbols

Regular Ideal Polytopes

§6.6 Fundamental Domains

Fundamental Regions

Locally Finite Fundamental Regions

Rigid Metric Spaces

Untitled

Dirichlet Domains

§6.7 Convex Fundamental Polyhedra

Fundamental Polyhedra

§6.8 Tessellations

Side-Pairing

Cycles of Polyhedra

Cycle Relations

§6.9 Historical Notes

CHAPTER 7: Classical Discrete Groups

§7.1 Reflection Groups

Coxeter Groups

Coxeter Graphs

Enhanced Coxeter Graphs

§7.2 Simplex Reflection Groups

Spherical Triangle Reection Groups

Euclidean Triangle Reection Groups

Hyperbolic Triangle Reection Groups

Barycentric Subdivision

Tetrahedron Reection Groups

Bilinear Forms

Classification of Simplex Reection Groups

§7.3 Generalized Simplex Reflection Groups

§7.4 The Volume of a Simplex

The Schläfli Differential Formula

§7.5 Crystallographic Groups

Bieberbach’s Theorem

The Splitting Group

Bieberbach Groups

§7.6 Torsion-Free Linear Groups

Valuation Rings

Selberg’s Lemma

§7.7 Historical Notes

CHAPTER 8: Geometric Manifolds

§8.1 Geometric Spaces

Free Group Actions

X-Space-Forms

§8.2 Clifford-Klein Space-Forms

Spherical Space-Forms

Euclidean Space-Forms

Hyperbolic Space-Forms

§8.3 (X,G)-Manifolds

X-Space-Forms

Metric (X;G)-Manifolds

§8.4 Developing

(X;G)-Maps

The Developing Map

Holonomy

§8.5 Completeness

Geodesic Completeness

Complete (X;G)-Manifolds

§8.6 Curvature

§8.7 Historical Notes

CHAPTER 9: Geometric Surfaces

§9.1 Compact Surfaces

Orientability

Surfaces-with-Boundary

§9.2 Gluing Surfaces

The Generalized Gluing Theorem

§9.3 The Gauss-Bonnet Theorem

§9.4 Moduli Spaces

Moduli Space

Teichmüller Space

The Dehn-Nielsen Theorem

Deformation Space

§9.5 Closed Euclidean Surfaces

§9.6 Closed Geodesics

Closed Curves

§9.7 Closed Hyperbolic Surfaces

Pairs of Pants

Teichmüller Space

§9.8 Hyperbolic Surfaces of Finite Area

Complete Gluing of Hyperbolic Surfaces

Cusps

Discrete Groups

§9.9 Historical Notes

CHAPTER 10: Hyperbolic 3-Manifolds

§10.1 Gluing 3-Manifolds

§10.2 Complete Gluing of 3-Manifolds

§10.3 Finite Volume Hyperbolic 3-Manifolds

The Whitehead Link Complement

The Borromean Rings Complement

§10.4 Hyperbolic Volume

Orthotetrahedra

The Lobachevsky Function

The Volume of an Orthotetrahedron

Ideal Tetrahedra

§10.5 Hyperbolic Dehn Surgery

Parameterization of Ideal Tetrahedra

Gluing Consistency Conditions

Hyperbolic Structures on the Figure-Eight Knot

The Uniqueness of the Complete Structure

The Metric Structure of the Link

Metric Completion

The Dehn Surgery Invariant

Dehn Surgery

§10.6 Historical Notes

CHAPTER 11: Hyperbolic n-manifolds

§11.1 Gluing n-Manifolds

Complete Gluing of n-Manifolds

§11.2 Poincaré’s Theorem

Poincaré’s Fundamental Polyhedron Theorem

§11.3 The Gauss-Bonnet Theorem

§11.4 Simplices of Maximum Volume

§11.5 Differential Forms

Tangent Maps

Euclidean Differential Forms

The Bundle of Skew-Symmetric k-Linear Functionals

Hyperbolic Differential Forms

The Integral of an n-Form

The Volume Form

The Integral of a k-Form over a k-Chain

§11.6 Simplicial Volume

Straight Singular k-Simplices

Haar Measure

§11.7 Measure Homology

The Measure Chain Complex

The de Rham Chain Complex

Straightening

Smearing

Representing the Fundamental Class

§11.8 Mostow Rigidity

Lipschitz Conditions

Pseudo-isometries

Measure Homology

Rigidity

§11.9 Historical Notes

CHAPTER 12: Geometrically Finite n-manifolds

§12.1 Limit Sets

§12.2 Limit Sets of Discrete Groups

The Ordinary Set

Nearest Point Retraction

Classical Schottky Groups

§12.3 Limit Points

Conical Limit Points

Cusped Limit Points

Cusp Points

§12.4 Geometrically Finite Discrete Groups

Geometrically Finite Convex Polyhedra

Horocusps

The Convex Core

Geometrically Finite Groups

§12.5 Nilpotent Groups

§12.6 The Margulis Lemma

Margulis Regions

Parabolic Fixed Points

Finiteness Properties of Geometrically Finite Groups

§12.7 Geometrically Finite Manifolds

Geometrically Finite Hyperbolic Manifolds

The Ideal Boundary of a Hyperbolic Manifold

Cusps

Finiteness Properties of Geometrically Finite Manifolds

§12.8 Arithmetic Hyperbolic Groups

Algebraic Integers

Number Fields

The Orthogonal Group of a Quadratic Form

Admissible Quadratic Forms

Arithmetic Hyperbolic Groups of the Simplest Type

Haar Measure

Compactness Criteria

§12.9 Historical Notes

CHAPTER 13: Geometric Orbifolds

§13.1 Orbit Spaces

§13.2 (X,G)-Orbifold

Order of a Point

Two-Dimensional Compact Geometric Orbifolds

Metric (X;G)-Orbifolds

§13.3 Developing Orbifolds

(X;G)-Paths

Homotopic (X;G)-Paths

Fundamental Orbifold Group

Holonomy

Universal Orbifold Covering Space

The Developing Map

Complete (X;G)-Orbifolds

§13.4 Gluing Orbifolds

Complete Gluing of Orbifolds

§13.5 Poincaré’s Theorem

Poincaré’s Fundamental Polyhedron Theorem

§13.6 Historical Notes

Bibliography

Index

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