Real hyperbolic space

Hyperbolic manifolds

Every complete, connected, simply connected manifold of constant negative curvature −1 is isometric to the real hyperbolic space H. As a result, the universal cover of any closed manifoldM of constant negative curvature −1, which is to say, a hyperbolic manifold, is H. Thus, every such M can be written as H‍/‍Γ, where Γ is a torsion-freediscrete group of isometries on H. That is, Γ is a lattice in SO(n, 1).