|
In geometry, a group of isometries of hyperbolic space is called geometrically finite if it has a well-behaved fundamental domain. A hyperbolic manifold is called geometrically finite if it can be described in terms of geometrically finite groups. ==Geometrically finite polyhedra== A convex polyhedron ''C'' in hyperbolic space is called geometrically finite if its closure in the conformal compactification of hyperbolic space has the following property: *For each point ''x'' in , there is a neighborhood ''U'' such that all faces of meeting ''U'' also pass through ''x'' . For example, every polyhedron with a finite number of faces is geometrically finite. In hyperbolic space of dimension at most 2, every geometrically finite polyhedron has a finite number of sides, but there are geometrically finite polyhedra in dimensions 3 and above with infinitely many sides. For example, in Euclidean space R''n'' of dimension ''n''≥2 there is a polyhedron ''P'' with an infinite number of sides. The upper half plane model of ''n''+1 dimensional hyperbolic space in R''n''+1 projects to R''n'', and the inverse image of ''P'' under this projection is a geometrically finite polyhedron with an infinite number of sides. A geometrically finite polyhedron has only a finite number of cusps, and all but finitely many sides meet one of the cusps. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Geometric finiteness」の詳細全文を読む スポンサード リンク
|