Gieseking manifold
In mathematics, the Gieseking manifold is a cusped hyperbolic 3-manifold of finite volume. It is non-orientable and has the smallest volume among non-compact hyperbolic manifolds, having volume approximately . It was discovered by Hugo Gieseking (1912). The volume is called Gieseking constant and has a closed-form,
where is the Clausen function. Similarly, Catalan's constant may also be expressed in terms of the Clausen function,
and also manifests as a volume. The Gieseking manifold can be constructed by removing the vertices from a tetrahedron, then gluing the faces together in pairs using affine-linear maps. Label the vertices 0, 1, 2, 3. Glue the face with vertices 0, 1, 2 to the face with vertices 3, 1, 0 in that order. Glue the face 0, 2, 3 to the face 3, 2, 1 in that order. In the hyperbolic structure of the Gieseking manifold, this ideal tetrahedron is the canonical polyhedral decomposition of David B. A. Epstein and Robert C. Penner. Moreover, the angle made by the faces is . The triangulation has one tetrahedron, two faces, one edge and no vertices, so all the edges of the original tetrahedron are glued together.
The Gieseking manifold has a
The Gieseking manifold is a fiber bundle over the circle with fiber the
See also
References
- Gieseking, Hugo (1912), Analytische Untersuchungen über Topologische Gruppen, Thesis, Muenster, JFM 43.0202.03
- MR 0894423
- MR 0918457.