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A 3-manifold is hyperbolic if it satisfies the following (equivalent) conditions:
- it admits a complete Riemannian metric of sectional curvature constant -1,
- it admits a Riemannian metric such that its universal covering (with the pull-back metric) is isometric to hyperbolic 3-space ,
- it is homeomorphic to , for some discrete, torsion-free group of isometries of hyperbolic 3-space.
Here, hyperbolic 3-space is the simply connected, complete, Riemannian manifold with sectional curvature constant -1. By Cartans Theorem, is unique up to isometry. There are different models for , like the upper half-space model, the Poincaré disc model or the hyperboloid model.
The ideal boundary can be identified with projective space . Isometries of hyperbolic 3-space act as conformal automorphisms of the ideal boundary. Thus we can identify the isometry group with the group of conformal automorphisms .
The group acts on the hyperboloid model and one can use this action to identify with the index two subgroup of orientation-preserving isometries. The action is transitive and has as a point stabilizer, thus is isometric to the homogeneous space .
The group acts by fractional-linear automorphisms on . This action on uniquely extends to an action on by orientation-preserving isometries. One can use this action to identify with . The action is transitive and has as a point stabilizer, thus is isometric to the homogeneous space .
Thus, if is oriented, then there are two more equivalent conditions:
An oriented 3-manifold is hyperbolic if and only if
- it is homeomorphic to for some discrete, torsion-free subgroup ,
- it is homeomorphic to for some discrete, torsion-free subgroup .
2 Construction and examples
5 Further discussion