Hyperbolic metric

In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension. They are especially studied in dimensions 2 and 3, where they are called hyperbolic surfaces and hyperbolic 3-manifolds, respectively. In these dimensions, they are important because most manifolds can be made into a hyperbolic manifold by a homeomorphism. This is a consequence of the uniformization theorem for surfaces and the geometrization theorem for 3-manifolds proved by Perelman.

A hyperbolic ${\displaystyle n}$-manifold is a complete Riemannian ${\displaystyle n}$-manifold of constant sectional curvature ${\displaystyle -1}$.

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