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.

with an explicitly written Riemannian metric; such constructions are referred to as models.

Sometimes the qualificative "real" is added to distinguish it from complex hyperbolic spaces.

-dimensional complete Riemannian manifold with a constant negative sectional curvature equal to −1.

[1] The unicity means that any two Riemannian manifolds that satisfy these properties are isometric to each other.

To prove the existence of such a space as described above one can explicitly construct it, for example as an open subset of

There are many such constructions or models of hyperbolic space, each suited to different aspects of its study.

Here is a list of the better-known models which are described in more detail in their namesake articles: 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): It is then a theorem that there are infinitely many such lines through P. This axiom still does not uniquely characterize the hyperbolic plane up to isometry; there is an extra constant, the curvature K < 0, that must be specified.

However, it does uniquely characterize it up to homothety, meaning up to bijections that only change the notion of distance by an overall constant.

By choosing an appropriate length scale, one can thus assume, without loss of generality, that K = −1.

The hyperbolic plane cannot be isometrically embedded into Euclidean 3-space by Hilbert's theorem.

The hyperbolic space also satisfies a linear isoperimetric inequality, that is there exists a constant

This is to be contrasted with Euclidean space where the isoperimetric inequality is quadratic.

Thus, every such M can be written as Hn‍/‍Γ, where Γ is a torsion-free discrete group of isometries on Hn.

According to the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic.

The quotient space H2‍/‍Γ of the upper half-plane modulo the fundamental group is known as the Fuchsian model of the hyperbolic surface.

The Poincaré half plane is also hyperbolic, but is simply connected and noncompact.

The analogous construction for three-dimensional hyperbolic surfaces is the Kleinian model.