Hyperbolic manifold

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 n-manifold of constant sectional curvature -1.

Every complete, connected, simply-connected manifold of constant negative curvature −1 is isometric to the real hyperbolic space ${\displaystyle \mathbb {H} ^{n}}$. As a result, the universal cover of any closed manifold M of constant negative curvature −1 is ${\displaystyle \mathbb {H} ^{n}}$. Thus, every such M can be written as ${\displaystyle \mathbb {H} ^{n}/\Gamma }$ where Γ is a torsion-free discrete group of isometries on ${\displaystyle \mathbb {H} ^{n}}$. That is, Γ is a discrete subgroup of ${\displaystyle SO_{1,n}^{+}\mathbb {R} }$. The manifold has finite volume if and only if Γ is a lattice.

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