# 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 perspective projection of a dodecahedral tessellation in H3. This is an example of what an observer might see inside a hyperbolic 3-manifold.
The Pseudosphere. Each half of this shape is a hyperbolic 2-manifold (i.e. surface) with boundary.

## Rigorous Definition

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

Its thick-thin decomposition has a thin part consisting of tubular neighborhoods of closed geodesics and ends which are the product of a Euclidean n-1-manifold and the closed half-ray. The manifold is of finite volume if and only if its thick part is compact.

For n>2 the hyperbolic structure on a finite volume hyperbolic n-manifold is unique by Mostow rigidity and so geometric invariants are in fact topological invariants.