Hyperbolic manifold

In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension.

This is a consequence of the uniformization theorem for surfaces and the geometrization theorem for 3-manifolds proved by Perelman.

Every complete, connected, simply-connected manifold of constant negative curvature

is isometric to the real hyperbolic space

As a result, the universal cover of any closed manifold

is a torsion-free discrete group of isometries on

Its thick–thin decomposition has a thin part consisting of tubular neighborhoods of closed geodesics and ends which are the product of a Euclidean (

The manifold is of finite volume if and only if its thick part is compact.

A simple non-trivial example, however, is the once-punctured torus.

This can be formed by taking an ideal rectangle in

– that is, a rectangle where the vertices are on the boundary at infinity, and thus don't exist in the resulting manifold – and identifying opposite images.

In a similar fashion, we can construct the thrice-punctured sphere, shown below, by gluing two ideal triangles together.

This also shows how to draw curves on the surface – the black line in the diagram becomes the closed curve when the green edges are glued together.

As we are working with a punctured sphere, the colored circles in the surface – including their boundaries – are not part of the surface, and hence are represented in the diagram as ideal vertices.

is a hyperbolic 3-manifold of finite volume.

-manifold is unique by Mostow rigidity and so geometric invariants are in fact topological invariants.

One of these geometric invariants used as a topological invariant is the hyperbolic volume of a knot or link complement, which can allow us to distinguish two knots from each other by studying the geometry of their respective manifolds.

A perspective projection of a dodecahedral tessellation in H 3 . 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.
(Left) A gluing diagram for the thrice-punctured sphere. Edges that are colored the same are glued together. Notice that the points where the lines meet (including the point at infinity) lie on the boundary of hyperbolic space, and so are not part of the surface. (Right) The surface glued together.