 in mathematics, hyperbolic space is a homogeneous space that can be characterized by a constant negative curvature, where in this case the curvature is the sectional curvature. it is the model of hyperbolic geometry. it is possible in dimensions 2 or higher, and is distinguished from euclidean spaces with zero curvature that define the euclidean geometry, and models of elliptic geometry that have a constant positive curvature. when embedded to a euclidean space (of a higher dimension), every point of a hyperbolic space is a saddle point. another distinctive property is the amount of space covered by the n-ball in hyperbolic n-space: it increases exponentially with respect to the radius of the ball, rather than polynomially. ==formal definition== hyperbolic n-space, denoted hn, is the maximally symmetric, simply connected, n-dimensional riemannian manifold with constant sectional curvature â1. 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. although hyperbolic space hn is diffeomorphic to rn its negative-curvature metric gives it very different geometric properties. hyperbolic 2-space, h2, is also called the hyperbolic plane. ==models of hyperbolic space== hyperbolic space, developed independently by nikolai lobachevsky and jÃ¡nos bolyai, is a geometrical space analogous to euclidean space, but such that euclid's parallel postulate is no longer assumed to hold. instead, the parallel postulate is replaced by the following alternative (in two dimensions): * given any line l and point p not on l, there are at least two distinct lines passing through p which do not intersect l. it is then a theorem that there are in fact infinitely many such lines through p. note that this axiom still does not uniquely characterize the hyperbolic plane up to isometry; there is an extra constant, the curvature , which must be specified. however, it does uniquely characterize it up to homothety, meaning up to bijections which only change the notion of distance by an overall constant. by choosing an appropriate length scale, one can thus assume, without loss of generality, that . hyperbolic spaces are constructed in order to m
Homogeneous_spaces Hyperbolic_geometry Topological_spaces