In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant negative sectional curvature, often taken to be −1 for simplicity. 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 R n {\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 distinguish it from complex hyperbolic spaces. 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.
Formal definition and models
Definition The n {\displaystyle n} -dimensional hyperbolic space or hyperbolic n {\displaystyle n} -space, usually denoted H n {\displaystyle \mathbb {H} ^{n}} , is the unique simply connected, n {\displaystyle n} -dimensional complete Riemannian manifold with a constant negative sectional curvature equal to −1. The unicity means that any two Riemannian manifolds that satisfy these properties are isometric to each other. It is a consequence of the Killing–Hopf theorem.
Models of hyperbolic space To prove the existence of such a space as described above one can explicitly construct it, for example as an open subset of R n {\displaystyle \mathbb {R} ^{n}} with a Riemannian metric given by a simple formula. There are many such constructions or models of hyperbolic space, each suited to different aspects of its study. They are isometric to each other according to the previous paragraph, and in each case an explicit isometry can be explicitly given. Here is a list of the better-known models which are described in more detail in their namesake articles:
Poincaré half-space model: this is the upper-half space { ( x 1 , … , x n ) ∈ R n : x n > 0 } {\displaystyle \{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}:x_{n}>0\}} with the metric d x 1 2 + ⋯ + d x n 2 x n 2 {\displaystyle {\tfrac {dx_{1}^{2}+\cdots +dx_{n}^{2}}{x_{n}^{2}}}}
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