
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a nonempty 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 (nonidentity elements act nontrivially), 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. ==Formal definition== Let ''X'' be a nonempty set and ''G'' a group. Then ''X'' is called a ''G''space if it is equipped with an action of ''G'' on ''X''.〔We assume that the action is on the ''left''. The distinction is only important in the description of ''X'' as a coset space.〕 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. Succinctly, if ''X'' is an object of the category C, then the structure of a ''G''space is a homomorphism: :$\backslash rho\; :\; G\; \backslash to\; \backslash mathrm\_\{\backslash mathbf\{C\}\}(X)$ into the group of automorphisms of the object ''X'' in the category C. The pair (''X'', ρ) defines a homogeneous space provided ρ(''G'') is a transitive group of symmetries of the underlying set of ''X''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「homogeneous space」の詳細全文を読む スポンサード リンク
