In mathematics, a homogeneous space is, very informally, a space that looks the same everywhere, as you move through it, with movement given by the action...
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In mathematics, a principal homogeneous space, or torsor, for a group G is a homogeneous space X for G in which the stabilizer subgroup of every point...
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codomain are vector spaces over a field F: a function f : V → W {\displaystyle f:V\to W} between two F-vector spaces is homogeneous of degree k {\displaystyle...
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Generalized flag variety (redirect from Projective homogeneous variety)
variety (or simply flag variety) is a homogeneous space whose points are flags in a finite-dimensional vector space V over a field F. When F is the real...
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Maurer–Cartan form (section On a homogeneous space)
symmetry on a space, where the symmetries of the space were transformations forming a Lie group. The geometries of interest were homogeneous spaces G/H, but...
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function defined by a homogeneous polynomial. A binary form is a form in two variables. A form is also a function defined on a vector space, which may be expressed...
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{\displaystyle k} -frames) are still homogeneous spaces for the orthogonal group, but not principal homogeneous spaces: any k {\displaystyle k} -frame can...
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Stiefel manifold (category Homogeneous spaces)
F n ) {\displaystyle V_{k}(\mathbb {F} ^{n})} can be viewed as a homogeneous space for the action of a classical group in a natural manner. Every orthogonal...
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projective space being considered. For example, two homogeneous coordinates are required to specify a point on the projective line and three homogeneous coordinates...
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Thus any symmetric space is a reductive homogeneous space, but there are many reductive homogeneous spaces which are not symmetric spaces. The key feature...
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Erlangen program (category Homogeneous spaces)
deeper and more general). In other words, the "traditional spaces" are homogeneous spaces; but not for a uniquely determined group. Changing the group...
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Lagrangian Grassmannian (category Topology of homogeneous spaces)
symplectic vector space V. Its dimension is 1/2n(n + 1) (where the dimension of V is 2n). It may be identified with the homogeneous space U(n)/O(n), where...
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system. The displacement vectors for that affine space are the solutions of the corresponding homogeneous linear system, which is a linear subspace. Linear...
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Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's Elements, it was the three-dimensional...
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{\mathcal {H}}} . Thus, anti-de Sitter is a reductive homogeneous space, and a non-Riemannian symmetric space. A d S n {\displaystyle \mathrm {AdS} _{n}} is...
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space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to −1. It is homogeneous...
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Homogeneity (disambiguation) (redirect from Homogeneous (mathematics))
of Leibniz Homogeneous space for a Lie group G, or more general transformation group Homogeneous function Homogeneous polynomial Homogeneous equation (linear...
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basis of a vector space often used to study the extrinsic differential geometry of smooth manifolds embedded in a homogeneous space. In lay terms, a frame...
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(orthonormal k-frames) are still homogeneous spaces for the orthogonal group, but not principal homogeneous spaces: any k-frame can be taken to any other...
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The geometry of a n-dimensional space can also be described with Riemannian geometry. An isotropic and homogeneous space can be described by the metric:...
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Fiber bundle (redirect from Base space)
group G {\displaystyle G} is given, so that each fiber is a principal homogeneous space. The bundle is often specified along with the group by referring to...
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the dual space, a homogeneous space for SU(2) and SL(2,C). Irreducible compact Hermitian symmetric spaces are exactly the homogeneous spaces of simple...
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Grassmannian (category Algebraic homogeneous spaces)
giving the Grassmannian a geometric structure is to express it as a homogeneous space. First, recall that the general linear group G L ( V ) {\displaystyle...
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Lorentz group (redirect from Homogeneous Lorentz group)
timelike vector, so the homogeneous space SO+(1, 3) / SO(3) is the momentum space of a massive particle; geometrically, this space is none other than three-dimensional...
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even-dimensional ones cannot. Complex projective space is a special case of a Grassmannian, and is a homogeneous space for various Lie groups. It is a Kähler manifold...
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Function space G-space Geometric space Green space (topological space) Hardy space Hausdorff space Heisenberg space Hilbert space Homogeneous space Inner...
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Equivalence class (redirect from Factor space)
set. Examples include quotient spaces in linear algebra, quotient spaces in topology, quotient groups, homogeneous spaces, quotient rings, quotient monoids...
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vector space. For a given n the elements of V n {\displaystyle V_{n}} are then called homogeneous elements of degree n. Graded vector spaces are common...
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Cartan connection (redirect from Cartan space)
Cartan connections describe the geometry of manifolds modelled on homogeneous spaces. The theory of Cartan connections was developed by Élie Cartan, as...
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Poisson point process (redirect from Non-homogeneous Poisson process)
the Poisson process located in some region of space. The resulting point process is called a homogeneous or stationary Poisson point process. In the second...
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