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.

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

References

Hyperbolic manifold Wikipedia

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