This book contains a systematic and comprehensive exposition of Lobachevskian geometry and the theory of discrete groups of motions in Euclidean space and Lobachevsky space. It is divided into two closely related parts: the first treats the geometry of spaces of constant curvature and the second discrete groups of motions of these. The authors give a very clear account of their subject describing it from the viewpoints of elementary geometry, Riemannian geometry and group theory. The result is a book which has no rival in the literature. Part I contains the classification of motions in spaces of constant curvature and non-traditional topics like the theory of acute-angled polyhedra and methods for computing volumes of non-Euclidean polyhedra. Part Ⅱ includes the theory of cristallographic, Fuchsian, and Kleinian groups and an exposition of Thurston\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s theory of deformations. The greater part of the book is accessible to first-year students in mathematics. At the same time the book includes very recent results which will be of interest to researchers in this field.
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目录
I. Geometry of Spaces of Constant Curvature Preface Chapter 1 Basic Structures §1. Definition of Spaces of Constant Curvature 1.1. Lie Groups of Transformations 1.2. Groups of Motions of a Riemannian Manifold 1.3. Invariant Riemannian Metrics on Homogeneous Spaces 1.4. Spaces of Constant Curvature 1.5. Three Spaces 1.6. Subspaces of the Space R(n,1) §2. The Classification Theorem 2.1. Statement of the Theorem 2.2. Reduction to Lie Algebras 2.3. The Symmetry 2.4. Structure of the Tangent Algebra of the Group of Motions 2.5. Riemann Space §3. Subspaces and Convexity 3.1. Involutions 3.2. Planes 3.3. Half-Spaces and Convex Sets 3.4. Orthogonal Planes §4. Metric 4.1. General Properties 4.2. Formulae for Distance in the Vector Model 4.3. Convexity of Distance Chapter 2 Models of Lobachevskij Space §1. Projective Models 1.1. Homogeneous Domains 1.2. Projective Model of Lobachevskij Space 1.3. Projective Euclidean Models. The Klein Model 1.4. "Affine" Subgroup of the Group of Automorphisms of a Quadric 1.5. Riemannian Metric and Distance Between Points in the Projective Model §2. Conformal Models 2.1. Conformal Space 2.2. Conformal Model of the Lobachevskij Space 2.3. Conformal Euclidean Models 2.4. Complex Structure of the Lobachevskij Plane §3. Matrix Models of the Spaces Л2 and Л3 3.1. Matrix Model of the Space Л2 3.2. Matrix Model of the Space Л3 Chapter 3 Plane Geometry §1. Lines 1.1. Divergent and Parallel Lines on the Lobachevskij Plane 1.2. Distance Between Points of Different Lines §2. Polygons 2.1. Definitions. Conditions of Convexity 2.2. Elementary Properties of Triangles 2.3. Polar Triangles on the Sphere 2.4. The Sum of the Angles in a Triangle 2.5. Existence of a Convex Polygon with Given Angles 2.6. The Angular Excess and the Area of a Polygon §3. Metric Relations 3.1. The Cosine Law for a Triangle 3.2. Other Relations in a Triangle 3.3. The Angle of Parallelism 3.4. Relations in Quadrilaterals, Pentagons, and Hexagons 3.5. The Circumference and the Area of a Disk §4. Motions and Homogeneous Lines 4.1. Classification of Motions of Two-Dimensional Spaces of Constant Curvature 4.2. Characterization of Motions of the Lobachevskij Plane in the Poincaré Model in Terms of Traces of Matrices 4.3. One-Parameter Groups of Motions of the Lobachevskij Plane and Their Orbits 4.4. The Form of the Metric in Coordinates Related to a Pencil Chapter 4 Planes, Spheres, Horospheres and Equidistant Surfaces §1. Relative Position of Planes 1.1. Pairs of Subspaces of a Euclidean Vector Space 1.2. Some General Notions 1.3. Pairs of Planes on the Sphere 1.4. Pairs of Planes in the Euclidean Space 1.5. Pseudo-orthogonal Transformations 1.6. Pairs of Hyperbolic Subspaces of the Space Rn,1 1.7. Pairs of Planes in the Lobachevskij Space 1.8. Pairs of Lines in the Lobachevskij Space 1.9. Pairs of Hyperplanes §2. Standard Surfaces 2.1. Definitions and Basic Facts 2.2. Standard Hypersurfaces 2.3. Similarity of Standard Hypersurfaces 2.4. Intersection of Standard Hypersurfaces §3 Decompositions into Semi-direct Products 3.1. Spherical, Horospherical, and Equidistant Decompositions 3.2. Spherical-equidistant Decomposition Chapter 5 Motions §1. General Properties of Motions 1.1. Description of Motions 1.2. Continuation of a Plane Motion 1.3. Displacement Function §2. Classification of Motions 2.1. Motions of the Sphere 2.2. Motions of the Euclidean Space 2.3. Motions of the Lobachevskij Space 2.4. One-parameter Groups of Motions §3. Groups of Motions and Similarities 3.1. Some Basic Notions 3.2. Criterion for the Existence of a Fixed Point 3.3. Groups of Motions of the Sphere 3.4. Groups of Motions of the Euclidean Space 3.5. Groups of Similarities 3.6. Groups of Motions of the Lobachevskij Space Chapter 6 Acute-angled Polyhedra §1. Basic Properties of Acute-angled Polyhedra 1.1. General Information on Convex Polyhedra 1.2. The Gram Matrix of a Convex Polyhedron 1.3. Acute-angled Families of Half-spaces and Acute-angled Polyhedra 1.4. Acute-angled Polyhedra on the Sphere and in Euclidean Space 1.5. Simplicity of Acute-angled Polyhedra §2. Acute-angled Polyhedra in Lobachevskij Space 2.1. Description in Terms of Gram Matrices 2.2. Combinatorial Structure 2.3. Description in Terms of Dihedral Angles Chapter 7 Volumes §1. Volumes of Sectors and Wedges 1.1. Volumes of Sectors 1.2. Volume of a Hyperbolic Wedge 1.3. Volume of a Parabolic Wedge 1.4. Volume of an Elliptic Wedge §2. Volumes of Polyhedra 2.1. Volume of a Simplex as an Analytic Function of the Dihedral Angles 2.2 Volume Differential 2.3. Volume of an Even-dimensional Polyhedron. The Poincaré and Schlafli Formulae 2.4. Volume of an Even-dimensional Polyhedron, The Gauss-Bonnet Formula §3 Volumes of 3-dimensional Polyhedra 3.1. The Lobachevskij Function 3.2. Double-rectangular Tetrahedra 3.3. Volume of a Double-rectangular Tetrahedron. The Lobachevskij Formula 3.4. Volumes of Tetrahedra with Vertices at Infinity 3.5. Volume of a Pyramid with the Apex at Infinity Chapter 8 Spaces of Constant Curvature as Riemannian Manifolds 1.1. Exponential Mapping 1.2. Parallel Translation 1.3. Curvature 1.4. Totally Geodesic Submanifolds 1.5. Hypersurfaces 1.6. Projective Properties 1.7. Conformal Properties 1.8. Pseudo-Riemannian Spaces of Constant Curvature References II. Discrete Groups of Motions of Spaces of Constant Curvature Preface Chapter 1 Introduction §1. Basic Notions 1.1. Definition of Discrete Groups of Motions 1.2. Quotient Spaces 1.3. Fundamental Domains 1.4. The Dirichlet Domain 1.5. Commensurable Groups §2. Origins of Discrete Groups of Motions 2.1. Symmetry Groups 2.2. Arithmetic Groups 2.3. Fundamental Groups of Space Forms Chapter 2 Fundamental Domains §1. Description of a Discrete Group in Terms of a Fundamental Domain 1.1. Normalization of a Fundamental Domain 1.2. Generators 1.3. Defining Relations 1.4. Points of a Fundamental Domain at Infinity 1.5. The Existence of a Discrete Group with a Given Fundamental Polyhedron 1.6. Limitations of Poincaré's Method 1.7. Homogeneous Decompositions §2. Geometrically Finite Groups of Motions of Lobachevskij Space 2.1. The Limit Set of a Discrete Group of Motions 2.2. Statement of the Main Results 2.3. Some General Properties of Discrete Groups of Motions of Lobachevskij Space 2.4. Compactification of the Quotient Space of a Crystallographic Group 2.5. A Criterion of Geometrical Finiteness Chapter 3 Crystallographic Groups §1 The Schoenflies-Bieberbach Theorem §1. The Schoenflies-Bieberbach Theorem 1.1. Statement of the Main Theorem 1.2. Commutators of Orthogonal Transformations and Motions 1.3. Proof of the Main Theorem 1.4. Arbitrary Discrete Groups of Motions of the Euclidean Space §2. Classification of Crystallographic Groups 2.1. Cohomological Description 2.2. Abstract Structure 2.3. Classification Steps 2.4. The Finiteness Theorem 2.5. Bravais Types 2.6. Some Classification Results 2.7. Euclidean Space Forms §3. Homogeneous Decompositions of Euclidean Space 3.1. The Finiteness Theorem 3.2. Parallelohedra Chapter 4 Fuchsian Groups §1. Fuchsian Groups from the Topological Point of View 1.1. Ramified Coverings 1.2. Signature and Uniformization of a Surface with Signature. Planar Groups 1.3. Planar Groups of Finite Type 1.4 Algebraic Structure of Planar Groups §2. Geometry of Fuchsian Groups 2.1. Introducing a Metric 2.2. Nielsen's Domain for a Fuchsian Group §3. The Teichmiiller Space of a Fuchsian Group 3.1. The Teichmiiller Space and the Moduli of an Abstract Group 3.2. The Teichmiiller Space of the Fundamental Group of a Closed Surface 3.3. Fuchsian Groups and Riemann Surfaces 3.4. Extensions of Fuchsian Groups. Maximal Fuchsian Groups Chapter 5 Reflection Groups §1. Basic Notions and Theorems 1.1. Coxeter Polyhedra 1.2. Discrete Reflection Groups 1.3. Coxeter Schemes 1.4. Reflection Groups on the Sphere and in Euclidean Space §2. Reflection Groups in the Lobachevskij Space 2.1. General Properties 2.2. Crystallographic Reflection Groups in the Lobachevskij Plane and in the Three-dimensional Lobachevskij Space 2.3. Lannér and Quasi-Lanér Groups 2.4. Some Other Examples 2.5. Restrictions on Dimension §3. Regular Polyhedra and Honeycombs 3.1. The Symmetry Group and the Schlafli Symbol of a Regular Polyhedron 3.2. Classification of Regular Polyhedra 3.3. Honeycombs Chapter 6 Arithmetic Groups §1. Description of Arithmetic Discrete Groups of Motions of Lobachevskij Space 1.1. Arithmetic Groups of the Simplest Type 1.2. Quaternion Algebras 1.3. Arithmetic Fuchsian Groups 1.4. Arithmetic Groups of Motions of the Space Л3 1.5. Arithmetic Groups of Motions of the Space Лn for n≥4 §2. Reflective Arithmetic Groups 2.1. Application of Reflection Groups to the Study of Arithmetic Groups 2.2. Classification Problem for Reflective Quadratic Forms §3 Existence of Non-Arithmetic Groups 3.1. Arithmeticity Criterion for Reflection Groups 3.2. Existence of Non-Arithmetic Reflection Groups 3.3. Existence of Non-Arithmetic Crystallographic Groups in Any Dimension Chapter 7 Sociology of Discrete Groups in the Lobachevskij Spaces §1. Rigidity and Deformation 1.1. Theorem on Strong Rigidity 1.2. Deformations 1.3. Dehn-Thurston Surgery §2. Commensurable Groups 2.1. Commensurator 2.2. Millson's Property §3. Covolumes 3.1. The Set of Covolumes 3.2. Discrete Groups of Minimal Covolume References Author Index Subject Index
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