Hyperbolic space
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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
- <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 Rn,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 Rn,1. Explicitly, the tangent space to a point x ∈ Hn 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.Template:Math-stub