Hyperbolic space

In mathematics, hyperbolic n-space, denoted Hn, 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, H2, is also called the hyperbolic plane.

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 Rn,1 is identical to Rn+1 except that the metric is given by the quadratic form

${\displaystyle \langle x,x\rangle =-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}.}$

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

Hyperbolic space, Hn, is then given as a hyperboloid of revolution in Rn,1:

${\displaystyle H^{n}=\{x\in \mathbb {R} ^{n,1}\mid \langle x,x\rangle =-1{\mbox{ and }}x_{0}>0\}.}$

The condition x0 > 0 selects only the top sheet of the two-sheeted hyperboloid so that Hn is connected. The space Hn is formally called the real hyperbolic space of dimension n .

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

Although hyperbolic space Hn is diffeomorphic to Rn its negative curvature metric gives it very different geometric properties.

Symmetry

The group O(n,1) is the Lie group of ${\displaystyle (n+1)\times (n+1)}$ real matrices that preserve the bilinear form

${\displaystyle \langle x,y\rangle =-x_{0}y_{0}+x_{1}y_{1}+x_{2}y_{2}+\cdots +x_{n}y_{n}.}$

That is, O(n,1) is the group of isometries of Minkowski space Rn,1 fixing the origin. This group is sometimes called the (n+1)-dimensional Lorentz group. The subgroup which preserves the orientation of x0 is called the orthochronous Lorentz group, denoted O+(n,1).

The action of O+(n,1) on Rn,1 restricts to an action on Hn. This group clearly preserves the hyperbolic metric on Hn. In fact, O+(n,1) is the full isometry group of Hn. 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 Hn 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 Hn. Which is to say that Hn is a homogeneous space for the action of SO+(n,1). The isotropy group of the vector ${\displaystyle (1,0,\ldots ,0)}$ is a matrix of the form

${\displaystyle {\begin{pmatrix}1&0&\ldots &0\\0&&&\\\vdots &&A&\\0&&&\\\end{pmatrix}}}$

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

The bilinear form ${\displaystyle \langle \,,\,\rangle }$ 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 Hn. As a result, the universal cover of any closed manifold M of constant negative curvature −1 is Hn. Thus, every such M can be written as Hn/Γ where Γ is a discrete group of isometries on Hn. That is, Γ is a lattice in SO+(n,1).