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.
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The complete simply connected Riemannian manifold of dimension with constant sectional curvature is hyperbolic space. In the upper half-space model its metric is for . Its ideal boundary in dimension three is the Riemann sphere.