# All Science Fair Projects

## Science Fair Project Encyclopedia for Schools!

 Search    Browse    Forum  Coach    Links    Editor    Help    Tell-a-Friend    Encyclopedia    Dictionary

# Science Fair Project Encyclopedia

For information on any area of science that interests you,
enter a keyword (eg. scientific method, molecule, cloud, carbohydrate etc.).
Or else, you can start by choosing any of the categories below.

# Hyperbolic space

(Redirected from Hyperbolic 3-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

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

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:

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

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

The 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.

Last updated: 05-28-2005 19:24:54
03-10-2013 05:06:04