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

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

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