Hyperbolic metric space information

In mathematics, a hyperbolic metric space is a metric space satisfying certain metric relations (depending quantitatively on a nonnegative real number...

In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal...

contains submanifolds endowed with a Riemannian metric yielding hyperbolic geometry. Model spaces of hyperbolic geometry of low dimension, say 2 or 3, cannot...

a hyperbolic group, also known as a word hyperbolic group or Gromov hyperbolic group, is a finitely generated group equipped with a word metric satisfying...

Poincaré metric Saccheri quadrilateral Systolic geometry Uniform tilings in hyperbolic plane δ-hyperbolic space "Curvature of curves on the hyperbolic plane"...

examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a metaphorical, rather than physical, notion...

a normalization of the metric): in particular, it is a CAT(-1/4) space. Complex hyperbolic spaces are also the symmetric spaces associated with the Lie...

theory of metric spaces named after the mathematician Mikhail Gromov. The Gromov product can also be used to define δ-hyperbolic metric spaces in the sense...

geometry Hyperbolic group, a finitely generated group equipped with a word metric satisfying certain properties characteristic of hyperbolic geometry...

induced metric in this case is positive-definite, and each sheet is a copy of hyperbolic n-space. For a detailed proof, see Minkowski space § Geometry...

acylindrically hyperbolic group is a group admitting a non-elementary 'acylindrical' isometric action on some geodesic hyperbolic metric space. This notion...

simplest examples of Gromov hyperbolic spaces. A metric space X {\displaystyle X} is a real tree if it is a geodesic space where every triangle is a tripod...

δ-hyperbolic space. One of the most common uses equivalence classes of geodesic rays. Pick some point O {\displaystyle O} of a hyperbolic metric space X...

metric has been applied to Perron–Frobenius theory and to constructing Gromov hyperbolic spaces. Let Ω be a convex open domain in a Euclidean space that...

Pseudometrics also arise in the theory of hyperbolic complex manifolds: see Kobayashi metric. Every measure space ( Ω , A , μ ) {\displaystyle (\Omega ,{\mathcal...

Euclidean case, three points of a hyperbolic space of an arbitrary dimension always lie on the same plane. Hence planar hyperbolic triangles also describe triangles...

r > 0 . There is also a curious relation to a hyperbolic angle and the metric defined on Minkowski space. Just as two dimensional Euclidean geometry defines...

accelerometer), then the hyperbolic coordinates are often called Rindler coordinates with the corresponding Rindler metric. If the observer is located...

class of hyperbolic metric spaces due to Gromov. Gromov's proof is given below for the Poincaré unit disk; the properties of hyperbolic metric spaces are developed...

In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension. They are especially studied in...