A torus. The standard torus is homogeneous under its diffeomorphism and homeomorphism groups, and the flat torus is homogeneous under its diffeomorphism, homeomorphism, and isometry groups.
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 of a group. Homogeneous spaces occur in the theories of Lie groups, algebraic groups and topological groups. More precisely, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts transitively. The elements of G are called the symmetries of X. A special case of this is when the group G in question is the automorphism group of the space X – here "automorphism group" can mean isometry group, diffeomorphism group, or homeomorphism group. In this case, X is homogeneous if intuitively X looks locally the same at each point, either in the sense of isometry (rigid geometry), diffeomorphism (differential geometry), or homeomorphism (topology). Some authors insist that the action of G be faithful (non-identity elements act non-trivially), although the present article does not. Thus there is a group action of G on X that can be thought of as preserving some "geometric structure" on X, and making X into a single G-orbit.
In mathematics, a homogeneousspace is, very informally, a space that looks the same everywhere, as you move through it, with movement given by the action...
In mathematics, a principal homogeneousspace, or torsor, for a group G is a homogeneousspace X for G in which the stabilizer subgroup of every point...
variety (or simply flag variety) is a homogeneousspace whose points are flags in a finite-dimensional vector space V over a field F. When F is the real...
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...
projective space being considered. For example, two homogeneous coordinates are required to specify a point on the projective line and three homogeneous coordinates...
F n ) {\displaystyle V_{k}(\mathbb {F} ^{n})} can be viewed as a homogeneousspace for the action of a classical group in a natural manner. Every orthogonal...
{\displaystyle k} -frames) are still homogeneousspaces for the orthogonal group, but not principal homogeneousspaces: any k {\displaystyle k} -frame can...
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...
Thus any symmetric space is a reductive homogeneousspace, but there are many reductive homogeneousspaces which are not symmetric spaces. The key feature...
symplectic vector space V. Its dimension is 1/2n(n + 1) (where the dimension of V is 2n). It may be identified with the homogeneousspace U(n)/O(n), where...
system. The displacement vectors for that affine space are the solutions of the corresponding homogeneous linear system, which is a linear subspace. Linear...
The geometry of a n-dimensional space can also be described with Riemannian geometry. An isotropic and homogeneousspace can be described by the metric:...
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's Elements, it was the three-dimensional...
deeper and more general). In other words, the "traditional spaces" are homogeneousspaces; but not for a uniquely determined group. Changing the group...
space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to −1. It is homogeneous...
the dual space, a homogeneousspace for SU(2) and SL(2,C). Irreducible compact Hermitian symmetric spaces are exactly the homogeneousspaces of simple...
timelike vector, so the homogeneousspace SO+(1, 3) / SO(3) is the momentum space of a massive particle; geometrically, this space is none other than three-dimensional...
even-dimensional ones cannot. Complex projective space is a special case of a Grassmannian, and is a homogeneousspace for various Lie groups. It is a Kähler manifold...
(orthonormal k-frames) are still homogeneousspaces for the orthogonal group, but not principal homogeneousspaces: any k-frame can be taken to any other...
group G {\displaystyle G} is given, so that each fiber is a principal homogeneousspace. The bundle is often specified along with the group by referring to...
{HP} ^{n}} and is a closed manifold of (real) dimension 4n. It is a homogeneousspace for a Lie group action, in more than one way. The quaternionic projective...
giving the Grassmannian a geometric structure is to express it as a homogeneousspace. First, recall that the general linear group G L ( V ) {\displaystyle...
Cartan connections describe the geometry of manifolds modelled on homogeneousspaces. The theory of Cartan connections was developed by Élie Cartan, as...
set. Examples include quotient spaces in linear algebra, quotient spaces in topology, quotient groups, homogeneousspaces, quotient rings, quotient monoids...
basis of a vector space often used to study the extrinsic differential geometry of smooth manifolds embedded in a homogeneousspace. In lay terms, a frame...
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