Hyperbolic Space - Hyperbolic Manifolds

Hyperbolic Manifolds

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