Hyperbolic manifold

In mathematics, hyperbolic manifolds are Riemannian manifolds with constant negative sectional curvature. They play an important role in low-dimensional topology, in particular in Thurston Geometrisierungsprogramm.

Definition

A hyperbolic manifold is a complete Riemannian manifold with constant sectional curvature. ( A Riemannian metric with constant sectional curvature is called hyperbolic metric. A hyperbolic manifold is thus a manifold with a complete hyperbolic metric. )

Equivalent Definition 1: A hyperbolic manifold is a Riemannian manifold whose universal covering is isometric to the hyperbolic space.

Equivalent Definition 2: A hyperbolic manifold is a Riemannian manifold of the form, where the hyperbolic space and a discrete subgroup of the group of isometries of hyperbolic space is.