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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.

Hyperbolic 3-manifold
Margulis lemma
Hyperbolic space
Hyperbolization theorem
Normally hyperbolic invariant manifold

References

Kapovich, Michael (2009) [2001], Hyperbolic manifolds and discrete groups, Modern Birkhäuser Classics, Boston, MA: Birkhäuser Boston, doi:10.1007/978-0-8176-4913-5, ISBN 978-0-8176-4912-8, MR 1792613
Maclachlan, Colin; Reid, Alan W. (2003), The arithmetic of hyperbolic 3-manifolds, Graduate Texts in Mathematics 219, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98386-8, MR 1937957
Ratcliffe, John G. (2006) [1994], Foundations of hyperbolic manifolds, Graduate Texts in Mathematics 149 (2nd ed.), Berlin, New York: Springer-Verlag, doi:10.1007/978-0-387-47322-2, ISBN 978-0-387-33197-3, MR 2249478
Hyperbolic Voronoi diagrams made easy, Frank Nielsen