Concept
Hyperbolic space
Related concepts (26)
Hyperboloid model
In geometry, the hyperboloid model, also known as the Minkowski model after Hermann Minkowski, is a model of n-dimensional hyperbolic geometry in which points are represented by points on the forward sheet S+ of a two-sheeted hyperboloid in (n+1)-dimensional Minkowski space or by the displacement vectors from the origin to those points, and m-planes are represented by the intersections of (m+1)-planes passing through the origin in Minkowski space with S+ or by wedge products of m vectors.
Riemann sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value for infinity. With the Riemann model, the point is near to very large numbers, just as the point is near to very small numbers. The extended complex numbers are useful in complex analysis because they allow for division by zero in some circumstances, in a way that makes expressions such as well-behaved.
Poincaré half-plane model
In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H , together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry. Equivalently the Poincaré half-plane model is sometimes described as a complex plane where the imaginary part (the y coordinate mentioned above) is positive. The Poincaré half-plane model is named after Henri Poincaré, but it originated with Eugenio Beltrami who used it, along with the Klein model and the Poincaré disk model, to show that hyperbolic geometry was equiconsistent with Euclidean geometry.
Poincaré disk model
In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk that are orthogonal to the unit circle or diameters of the unit circle. The group of orientation preserving isometries of the disk model is given by the projective special unitary group PSU(1,1), the quotient of the special unitary group SU(1,1) by its center {I, −I}.
Eugenio Beltrami
Eugenio Beltrami (16 November 1835 – 18 February 1900) was an Italian mathematician notable for his work concerning differential geometry and mathematical physics. His work was noted especially for clarity of exposition. He was the first to prove consistency of non-Euclidean geometry by modeling it on a surface of constant curvature, the pseudosphere, and in the interior of an n-dimensional unit sphere, the so-called Beltrami–Klein model. He also developed singular value decomposition for matrices, which has been subsequently rediscovered several times.
Pseudo-Euclidean space
In mathematics and theoretical physics, a pseudo-Euclidean space is a finite-dimensional real n-space together with a non-degenerate quadratic form q. Such a quadratic form can, given a suitable choice of basis (e1, ..., en), be applied to a vector x = x1e1 + ⋯ + xnen, giving which is called the scalar square of the vector x. For Euclidean spaces, k = n, implying that the quadratic form is positive-definite. When 0 < k < n, q is an isotropic quadratic form, otherwise it is anisotropic.
Hyperbolic manifold
In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension. They are especially studied in dimensions 2 and 3, where they are called hyperbolic surfaces and hyperbolic 3-manifolds, respectively. In these dimensions, they are important because most manifolds can be made into a hyperbolic manifold by a homeomorphism. This is a consequence of the uniformization theorem for surfaces and the geometrization theorem for 3-manifolds proved by Perelman.
Symmetric space
In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, leading to consequences in the theory of holonomy; or algebraically through Lie theory, which allowed Cartan to give a complete classification. Symmetric spaces commonly occur in differential geometry, representation theory and harmonic analysis.
Discrete group
In mathematics, a topological group G is called a discrete group if there is no limit point in it (i.e., for each element in G, there is a neighborhood which only contains that element). Equivalently, the group G is discrete if and only if its identity is isolated. A subgroup H of a topological group G is a discrete subgroup if H is discrete when endowed with the subspace topology from G. In other words there is a neighbourhood of the identity in G containing no other element of H.
De Sitter space
In mathematical physics, n-dimensional de Sitter space (often abbreviated to dSn) is a maximally symmetric Lorentzian manifold with constant positive scalar curvature. It is the Lorentzian analogue of an n-sphere (with its canonical Riemannian metric). The main application of de Sitter space is its use in general relativity, where it serves as one of the simplest mathematical models of the universe consistent with the observed accelerating expansion of the universe.