双曲几何讲义

PrefaceChapter A.Hyperbolic Space A.1 Models for Hyperbolic Space A.2 Isometries of Hyperbolic Space； Hyperboloid Model A.3 Conformal Geometry A.4 Isometrics of Hyperbolic Space； Disc and Half-space Models A.5 Geodesics, Hyperbolic Subspaces and Miscellaneous Facts A.6 Curvature of Hyperbolic SpaceChapter B.Hyperbolic Manifolds and the Compact Two-dime
ional Case B.1 Hyperbolic, Elliptic and Flat Manifolds B.2 Topology of Compact Oriented Surfaces B.3 Hyperbolic, Elliptic and Flat Surfaces B.4 TeichmiiUer SpaceChapter C.The Rigidity Theorem (Compact Case) C.1 Fi
t Step of the Proof； Exte
ion of Psettdo-isometries C.2 Second Step of the Proof； Volume of Ideal Simplices C.3 Gromov Norm of a Compact Manifold C.4 Third Step of the Proof；the Gromov Norm and the Vohtme Are Proportional C.5 Conclusion of the Proof, Corollaries and Generalizatio
Chapter D.Margulis' Lemma and its Applicatio
D.1 Margulis' Lemma D.2 Local Geometry of a Hyperbolic Manifohl D.3 Ends of a Hyperbolic .ManifoldChapter E.The Space of Hyperbolic Manifolds and the Volume Function E.1 The Chabauty and the Geometric Topology E.2 Convergence in the Geometric Topology； Opening Cusps. The Case of Dime
ion at least Three E.3 The Case of Dime
ion Different from Three Conclusio
and Examples E.4 The Three-dime
ional Case； Josge
en's Part of the So-called Jorge
en-Thu
ton Theory E.5 The Three-dime
ional Case. Thu
ton's Hyperbolic Surgery Theorem； Statement and Preliminaries E.5-i Definition and Fi
t Properties of T3(Non-compact Three-manifolds with "Triangulation" Without Vertices) E.5-ii Hyperbolic Structures on an Element of T3 and Realization of the Complete Structure E.5-iii Elements of T3 and Standard Spines E.5-iv Some Links Whose Complements are Realized as Elements of T3 E.6 Proof of Thu
ton's Hyperbolic Surgery Theorem E.6-i Algebraic Equatio
of H(M) (Hyperbolic Structures Supported by M∈T3) E.6-ii Dime
ion of H(M)； General Case E.6-iii The Case M is Complete Hyperbolic； the Space of Deformatio
E.6-iv Completion of the Deformed Hyperbolic Structures and Conclusion of the Proof E.7 Applicatio
to the Study of the Volume Function and Complements about Three-dime
ional Hyperbolic GeometryChapter F. Bounded Cohomology, a Rough Outline F.1 Singudar Cohomology F.2 Bounded Singular Coliomology F.3 Flat Fiber Bundles F.4 Euler Class of a Flat Vector Bundle F.5 Flat Vector Bundles on Surfaces and the Milnor-Sullivan Theorem F.6 Sullivan's Conjecture and Amenable GroupsSubject IndexNotation IndexReferences

Riccardo Benedetti、Carlo Petronio所著的《双曲几何讲义》是一部讲述双曲几何的本科生教程，重点强调双曲流形上的几何。旨在为读者全面讲述基础结果，独立性强，完整，详尽，自成体系。在讲述双曲空间的经典材料和Teichmüller空间之后，接着以Mostow 刚性定理和Margulis定理这两个基本结论为核心展开讲述。这些形成了学习Chabauty和几何拓扑的基础；并且深入全面地剖析了Wang定理和 Jorgensen-Thurston 理论，给予讲述三维例子很大的空间；同时，以依附于理想四面体的三流形表示为基础，全面介绍了双曲手术定理。