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... > algebraic topology > manifold > differentiable manifold > Lie group > homogeneous space

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geometry > differential geometry > differentiable manifold > Lie group > homogeneous space

topology > differential topology > differentiable manifold > Lie group > homogeneous space

algebra > group theory > topological group > Lie group > homogeneous space

algebra > abstract algebra > algebraic structure > homogeneous space

homogeneous space

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 which can be thought of as preserving some "geometric structure" on X, and making X into a single G-orbit.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Homogeneous_space)

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