双曲几何讲义
2012-8
世界图书出版公司
Riccardo Benedetti,Carlo Petronio
330
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One of the main themes of this book is the conflict between
the "flexibility' and the "rigidity properties of the hyperbolic
manifolds： the first radical difference arises between the case of
dimension 2 and the case of higher dimensions （as proved in
chapters B and C）， an elementary feature of thus phenomenon being
the difference between the Riemann mapping theorem and Liouville's
theorem， as pointed out in chapter A. Thus chapter is rather
clementary and most of its material may' be the object of an
undergraduate course.
Together with the rigidity theorem， a basic tool for the study of
hyperbolic manifolds is Margulis' lemma， a detailed proof of which
we give in chapter D; as a consequence of this result in the same
chapter we also give a rather accurate description， in all
dimensions， of the thin-thick decomposition of a hyperbolic
manifold （especially in case of finite volume）.
preface
chapter a.hyperbolic space
a.1 models for hyperbolic space
a.2 isometries of hyperbolic space: hyperboloid model
a.3 conformal geometry
a.4 isometries of hyperbolic space: disc and half-space
models
a.5 geodesics, hyperbolic subspaces and misceuaneo,s facts
a.6 curvature of hyperbolic space
chapter b.hyperbolic manifolds and the compact two-dimensional
case
b.1 hyperbolic, elliptic and flat manifolds
b.2 topology of compact oriented surfaces
b.3 hyperbolic, elliptic and flat surfaces
b.4 teichmiiller space
chapter c.the rigidity theorem (compact case)
c.1 first step of the proof: extension of pseudo-isometrics
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 volume are
proportional
c.5 conclusion of the proof, corollaries and generalizations
chapter d.margulis' lemma and its applications
d.1 margnlis' lemma
d.2 local geometry of a hyperbolic manifold
d.3 ends of a hyperbolic manifold
chapter e.the space of hyperbolic manifolds and the volume
function
e.1 the chahauty and the geometric topology
e.2 convergence in the geometric topology: opening cusps.the case
of dimension at least three
e.3 the case of dimension different from three.conclusions and
examples
e.4 the three-dimensional case: jorgensen's part of the
so-calledjorgensen-tlmrston theory
e.5 the three-dimensional case. thurston's hyperbolic surgery
theorem: statement and preliminaries
e.5-i definition and first properties of ts (non-compact
three-manifolds with "triangulation" without vertices)
e.5-ii hyperbolic structures on an element of ts and realization
of the complete structure
e.5-iii elements of ts and standard spines
e.5-iv some links whose complements are realized a.s elements of
ts
e.6 proof of thurston's hyperbolic surgery theorem
e.6-i algebraic equations of h(m)(hyperbolic structures supported
by m∈t3)
e.6-ii dimension of h(m): general ca.se
e.6-iii the case m is complete hyperbolic:the space of
deformations
e.6-iv completion of the deformed hyperbolic structures and
conclusion of the proof
e.7 applications to the study of the volume f, mction and
complements about three-dimensional hyperbolic geometry
chapter f.bounded cohomology, a rough outline
f.1 singular cohomology
f.2 bounded singular cohomology
f.3 flat fiber bundles
f.4 euler class of a flat vector bundle
f.5 flat vector bundles on surfaces and the milner-sullivan
theorem
f.6 suuivan's conjecture and amenable groups
subject index
notation index
references
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