Hyperbolic manifold

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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[edit]

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 Hn. As a result, the universal cover of any closed manifold M of constant negative curvature −1 is Hn. Thus, every such M can be written as Hn/Γ where Γ is a torsion-free discrete group of isometries on Hn. 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.

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