Inhomogeneous space

In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, 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.

Let X be a non-empty set and G a group. Then X is called a G-space if it is equipped with an action of G on X. Note that automatically G acts by automorphisms (bijections) on the set. If X in addition belongs to some category, then the elements of G are assumed to act as automorphisms in the same category. Thus the maps on X effected by G are structure preserving. A homogeneous space is a G-space on which G acts transitively.

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