 Hyperbolic space

The complete simply connected Riemannian manifold of dimension $n$ with constant sectional curvature $-1$ is hyperbolic space. In the upper half-space model its metric is $d{s}^2=(d{x}_1^2+\cdots +d{x}_n^2)/x_n^2$ for $x_n>0$. Its ideal boundary in dimension three is the Riemann sphere.

Hyperbolic space is a type of non-Euclidean geometry that generalizes the concepts of traditional Euclidean geometry to a space with a constant negative curvature. In hyperbolic geometry, the parallel postulate of Euclidean geometry—specifically, that through a point not on a given line, there is exactly one line parallel to the given line—does not hold. Instead, through a point not on a given line, there are infinitely many lines that do not intersect the given line.