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

<math>\langle x, x\rangle = -x_0^2 + x_1^2 + x_2^2 + \cdots + x_n^2.<math>

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 R^n,1:

<math>H^n = \{x \in \mathbb{R}^{n,1} \mid \langle x, x\rangle = -1 \mbox{ and } x_0 > 0\}.<math>

The condition x₀ > 0 selects only the top sheet of the two-sheeted hyperboloid so that Hⁿ is connected.

The metric on Hⁿ is induced from the metric on R^n,1. Explicitly, the tangent space to a point x ∈ Hⁿ 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 Hⁿ is positive-definite even through the metric on R^n,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.