# Hyperbolic space

In mathematics, **hyperbolic n-space**, denoted

*H*

^{n}, is the maximally symmetric, simply connected,

*n*-dimensional Riemannian manifold with constant sectional curvature −1. Hyperbolic space is the principal example of a space exhibiting hyperbolic geometry. It can be thought of as the negative curvature analogue of the

*n*-sphere.

Hyperbolic 2-space, *H*^{2}, is also called the hyperbolic plane.

## Contents

## Definition

Hyperbolic space is most commonly defined as a submanifold of (*n*+1)-dimensional Minkowski space, in much the same manner as the *n*-sphere is defined as a submanifold of (*n*+1)-dimensional Euclidean space. Minkowski space **R**^{n,1} is identical to **R**^{n+1} except that the metric is given by the quadratic form

Note that the Minkowski metric is not positive-definite, but rather has signature (−, +, +, …, +). This gives it rather different properties than Euclidean space.

Hyperbolic space, *H*^{n}, is then given as a hyperboloid of revolution in **R**^{n,1}:

The condition *x*_{0} > 0 selects only the top sheet of the two-sheeted hyperboloid so that *H*^{n} is connected. The space *H*^{n} is formally called the **real hyperbolic space of dimension n **.

The metric on *H*^{n} is induced from the metric on **R**^{n,1}. Explicitly, the tangent space to a point *x* ∈ *H*^{n} can be identified with the orthogonal complement of *x* in **R**^{n,1}. The metric on the tangent space is obtained by simply restricting the metric on **R**^{n,1}. It is important to note that the metric on *H*^{n} is positive-definite even through the metric on **R**^{n,1} is not. This means that *H*^{n} is a true Riemannian manifold (as opposed to a pseudo-Riemannian manifold).

Although hyperbolic space *H*^{n} is diffeomorphic to **R**^{n} its negative curvature metric gives it very different geometric properties.

## Symmetry

The group O(*n*,1) is the Lie group of real matrices that preserve the bilinear form

That is, O(*n*,1) is the group of isometries of Minkowski space **R**^{n,1} fixing the origin. This group is sometimes called the (*n*+1)-dimensional Lorentz group. The subgroup which preserves the orientation of *x*_{0} is called the *orthochronous Lorentz group*, denoted O^{+}(*n*,1).

The action of O^{+}(*n*,1) on **R**^{n,1} restricts to an action on *H*^{n}. This group clearly preserves the hyperbolic metric on *H*^{n}. In fact, O^{+}(*n*,1) is the full isometry group of *H*^{n}. This isometry group has dimension *n*(*n*+1)/2, the maximal number of isometries for a Riemannian manifold. Therefore, hyperbolic space is said to be *maximally symmetric*. The group of orientation preserving isometries of *H*^{n} is the group SO^{+}(*n*,1), which is the identity component of the full Lorentz group.

The orientation preserving isometry group SO^{+}(*n*,1) acts transitively and faithfully on *H*^{n}. Which is to say that *H*^{n} is a homogeneous space for the action of SO^{+}(*n*,1). The isotropy group of the vector is a matrix of the form

where *A* is a matrix in the rotation group SO(*n*); that is, *A* is an orthogonal matrix with determinant +1. Hyperbolic space *H*^{n} is therefore isomorphic to the quotient space SO^{+}(*n*,1)/SO(*n*).

The bilinear form is the Cartan-Killing form, the unique SO^{+}(*n*,1)-invariant quadratic form on SO^{+}(*n*,1).

## General case

Every complete, simply-connected manifold of constant negative curvature −1 is isometric to the real hyperbolic space *H*^{n}. As a result, the universal cover of any closed manifold *M* of constant negative curvature −1 is *H*^{n}. Thus, every such *M* can be written as *H*^{n}/Γ where Γ is a discrete group of isometries on *H*^{n}. That is, Γ is a lattice in SO^{+}(*n*,1).