More precisely, looking at the quadratic form on , its restriction to the tangent spaces of the upper sheet of the hyperboloid given by are definite positive, hence they endow it with a Riemannian metric which turns out to be of constant curvature -1. The isometry to the previous models can be realised by stereographic projection from the hyperboloid to the plane , taking the vertex from which to project to be for the ball and a point at infinity in the cone inside projective space for the half-space. Klein model: This is another model realised on the unit ball of ; rather than being given as an explicit metric it is usually presented as obtained by using stereographic projection from the hyperboloid model in Minkowski space to its horizontal tangent plane (that is, ) from the origin . Symmetric space: Hyperbolic -space can be realised as the symmetric space of the simple Lie group (the group of isometries of the quadratic form with positive determinant); as a set the latter is the coset space . The isometry to the hyperboloid model is immediate through the action of the connected component of on the hyperboloid. Geometric properties Parallel lines Hyperbolic space, developed independently by Nikolai Lobachevsky, János Bolyai and Carl Friedrich Gauss, 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 infinitely many such lines through P. 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 . Euclidean embeddings The hyperbolic plane cannot be isometrically embedded into Euclidean 3-space by Hilbert's theorem. On the other hand the Nash embedding theorem implies that hyperbolic n-space can be isometrically embedded into some Euclidean space of larger dimension (4 for the hyperbolic plane). When isometrically embedded to a Euclidean space every