A perspective projection of a dodecahedral tessellation in H3. Four dodecahedra meet at each edge, and eight meet at each vertex, like the cubes of a cubic tessellation in E3
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. There are many ways to construct it as an open subset of with an explicitly written Riemannian metric; such constructions are referred to as models. Hyperbolic 2-space, H2, which was the first instance studied, is also called the hyperbolic plane.
It is also sometimes referred to as Lobachevsky space or Bolyai–Lobachevsky space after the names of the author who first published on the topic of hyperbolic geometry. Sometimes the qualificative "real" is added to differentiate it from complex hyperbolic spaces, quaternionic hyperbolic spaces and the octononic hyperbolic plane, which are the other symmetric spaces of negative curvature.
Hyperbolic space serves as the prototype of a Gromov hyperbolic space, which is a far-reaching notion including differential-geometric as well as more combinatorial spaces via a synthetic approach to negative curvature. Another generalisation is the notion of a CAT(−1) space.
^
Grigor'yan, Alexander; Noguchi, Masakazu (1998), "The heat kernel on hyperbolic space", The Bulletin of the London Mathematical Society, 30 (6): 643–650, doi:10.1112/S0024609398004780, MR 1642767
In mathematics, hyperbolicspace of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal...
In mathematics, a hyperbolic metric space is a metric space satisfying certain metric relations (depending quantitatively on a nonnegative real number...
2-dimensional (planar) hyperbolic geometry and the differences and similarities between Euclidean and hyperbolic geometry. See hyperbolicspace for more information...
In mathematics, hyperbolic complex space is a Hermitian manifold which is the equivalent of the real hyperbolicspace in the context of complex manifolds...
circular function Hyperbolic geometric graph, a random network generated by connecting nearby points sprinkled in a hyperbolicspaceHyperbolic geometry, a...
In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolicspace of some dimension. They are especially studied in...
This article lists the regular polytopes in Euclidean, spherical and hyperbolicspaces. This table shows a summary of regular polytope counts by rank. Only...
precisely in geometric group theory, a hyperbolic group, also known as a word hyperbolic group or Gromov hyperbolic group, is a finitely generated group...
non-Euclidean spaces such as hyperbolicspace. A typical sphere packing problem is to find an arrangement in which the spheres fill as much of the space as possible...
such as hyperbolicspace. A further classical example is the space of lines in projective space of three dimensions (equivalently, the space of two-dimensional...
actually imaginary, which turns rotations into rotations in hyperbolicspace (see hyperbolic rotation). This idea, which was mentioned only briefly by Poincaré...
set of hyperbolic uniform honeycombs. (more unsolved problems in mathematics) In hyperbolic geometry, a uniform honeycomb in hyperbolicspace is a uniform...
Euclidean case, three points of a hyperbolicspace of an arbitrary dimension always lie on the same plane. Hence planar hyperbolic triangles also describe triangles...
examples are Euclidean n-space, the n-dimensional sphere, and hyperbolicspace, although a space form need not be simply connected. The Killing–Hopf theorem...
stereographic projections into 3-space. Studies of non-Euclidean (hyperbolic and elliptic) and other spaces such as complex spaces, discovered over the preceding...
In geometry, hyperbolic motions are isometric automorphisms of a hyperbolicspace. Under composition of mappings, the hyperbolic motions form a continuous...
considered tessellations, or tilings, of spherical space. Tessellations of euclidean and hyperbolicspace may also be considered regular polytopes. Note that...
non-compact symmetric spaces is a generalization of the well known duality between spherical and hyperbolic geometry. A symmetric space with a compatible...
Global Information
