Hyperbolic space

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 ${\displaystyle \mathbb {R} ^{n}}$ 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.