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Concept
Hyperbolic manifold
Summary
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 hyperbolic -manifold is a complete Riemannian -manifold of constant sectional curvature .
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 of constant negative curvature is . Thus, every such can be written as where is a torsion-free discrete group of isometries on . That is, is a discrete subgroup of . The manifold has finite volume if and only if 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 ()-manifold and the closed half-ray. The manifold is of finite volume if and only if its thick part is compact.
The simplest example of a hyperbolic manifold is hyperbolic space, as each point in hyperbolic space has a neighborhood isometric to hyperbolic space.
A simple non-trivial example, however, is the once-punctured torus. This is an example of an (Isom(), )-manifold. 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.
Official source
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