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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.
An example of a noncompact, finite volume hyperbolic manifold obtained in this way is the Gieseking manifold which is constructed by gluing faces of a regular ideal hyperbolic tetrahedron together. It is also possible to construct a finite-volume, complete hyperbolic manifold when the gluing is not complete.
Then a (,) manifold is simply a flat manifold. A particularly interesting example is when is a Riemannian symmetric space, for example hyperbolic space. The simplest such example is the hyperbolic plane, whose isometry group is isomorphic to = ().
In mathematics, Seifert–Weber space (introduced by Herbert Seifert and Constantin Weber) is a closed hyperbolic 3-manifold. It is also known as Seifert–Weber dodecahedral space and hyperbolic dodecahedral space. It is one of the first discovered examples of closed hyperbolic 3-manifolds.
In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to −1. [1] It is homogeneous , and satisfies the stronger property of being a symmetric space .
A hyperbolic manifold is called geometrically finite if it has a finite number of components, each of which is the quotient of hyperbolic space by a geometrically finite discrete group of isometries (Ratcliffe 1994, 12.7).
The Weeks manifold is the hyperbolic three-manifold of smallest volume [3] and the Meyerhoff manifold is the one of next smallest volume. The complement in the three-sphere of the figure-eight knot is an arithmetic hyperbolic three-manifold [4] and attains the smallest volume among all cusped hyperbolic three-manifolds. [5]
It is also known as Seifert–Weber dodecahedral space and hyperbolic dodecahedral space. It is one of the first discovered examples of closed hyperbolic 3-manifolds. It is constructed by gluing each face of a dodecahedron to its opposite in a way that produces a closed 3-manifold. There are three ways to do this gluing consistently.