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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.
In mathematics, a hyperbolic metric space is a metric space satisfying certain metric relations (depending quantitatively on a nonnegative real number δ) between points. The definition, introduced by Mikhael Gromov , generalizes the metric properties of classical hyperbolic geometry and of trees .
However, the entire hyperbolic plane cannot be embedded into Euclidean space in this way, and various other models are more convenient for abstractly exploring hyperbolic geometry. There are four models commonly used for hyperbolic geometry: the Klein model , the Poincaré disk model , the Poincaré half-plane model , and the Lorentz or ...
In mathematics, a space form is a complete Riemannian manifold M of constant sectional curvature K. The three most fundamental examples are Euclidean n -space , the n -dimensional sphere , and hyperbolic space , although a space form need not be simply connected .
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.
If the signature (−, +, +) is chosen, then the scalar square of chords between distinct points on the same sheet of the hyperboloid will be positive, which more closely aligns with conventional definitions and expectations in mathematics. Then n-dimensional hyperbolic space is a Riemannian space and distance or length can be defined as the ...
The (6,4,2) triangular hyperbolic tiling. The triangle group corresponding to this tiling has a circle as its Gromov boundary. In mathematics, the Gromov boundary of a δ-hyperbolic space (especially a hyperbolic group) is an abstract concept generalizing the boundary sphere of hyperbolic space.
This is useful to produce actions of hyperbolic groups on real trees. Such actions are analyzed using the so-called Rips machine. A case of particular interest is the study of degeneration of groups acting properly discontinuously on a real hyperbolic space (this predates Rips', Bestvina's and Paulin's work and is due to J. Morgan and P. Shalen ...