Definition

The -dimensional hyperbolic space or hyperbolic -space, usually denoted , is the unique simply connected, -dimensional complete Riemannian manifold with a constant negative sectional curvature equal to -1. The unicity means that any two Riemannian manifolds that satisfy these properties are isometric to each other. It is a consequence of the Killing-Hopf theorem.

Models of hyperbolic space

To prove the existence

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