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Hyperbolic space, developed independently by Nikolai Lobachevsky, János Bolyai and Carl Friedrich Gauss, is a geometric space analogous to Euclidean space, but such that Euclid's parallel postulate is no longer assumed to hold. Instead, the parallel postulate is replaced by the following alternative (in two dimensions):
The definition of an hyperbolic space in terms of the Gromov product can be seen as saying that the metric relations between any four points are the same as they would be in a tree, up to the additive constant . More generally the following property shows that any finite subset of an hyperbolic space looks like a finite tree.
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.
Hyperbolic geometry is generally introduced in terms of the geodesics and their intersections on the hyperbolic plane. [ 38 ] Once we choose a coordinate chart (one of the "models"), we can always embed it in a Euclidean space of same dimension, but the embedding is clearly not isometric (since the curvature of Euclidean space is 0).
Then n-dimensional hyperbolic space is a Riemannian space and distance or length can be defined as the square root of the scalar square. If the signature (+, −, −) is chosen, scalar square between distinct points on the hyperboloid will be negative, so various definitions of basic terms must be adjusted, which can be inconvenient.
The solutions of hyperbolic equations are "wave-like". If a disturbance is made in the initial data of a hyperbolic differential equation, then not every point of space feels the disturbance at once. Relative to a fixed time coordinate, disturbances have a finite propagation speed. They travel along the characteristics of the equation.
Hyperbolic motions are often taken from inversive geometry: these are mappings composed of reflections in a line or a circle (or in a hyperplane or a hypersphere for hyperbolic spaces of more than two dimensions). To distinguish the hyperbolic motions, a particular line or circle is taken as the absolute.
A group G is relatively hyperbolic with respect to a subgroup H if, after contracting the Cayley graph of G along H-cosets, the resulting graph equipped with the usual graph metric becomes a δ-hyperbolic space and, moreover, it satisfies a technical condition which implies that quasi-geodesics with common endpoints travel through approximately the same collection of cosets and enter and exit ...