Some applications to hyperbolic geometry

Abstract

Consider a compact, connected, orientable 3-manifold with a hyperbolic metric. By Mostow’s Rigidity Theorem [14, Ch. C], the hyperbolic metric is unique up to isometry. Hence the geometric invariants of the metric, such as the volume or the injectivity radius, are topological invariants. The recent proof of Thurston’s Geometrization Conjecture based on the results by Perelman [78], [79], [80], has increased greatly the interest in hyperbolic 3-manifolds. Let us remark that on a compact Riemann surface of genus g ≥ 2, the space of hyperbolic metrics up to isotopy is the Teichmüller space, a real analytic manifold of real dimension 6g−6; see [14, Ch. B].KeywordsHyperbolic ManifoldCompact Riemannian ManifoldHyperbolic GeometryInjectivity RadiusHeegaard SplittingThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.