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In mathematics, a Klein geometry is a type of geometry motivated by Felix Klein in his influential Erlangen program. More specifically, it is a homogeneous space ''X'' together with a transitive action on ''X'' by a Lie group ''G'', which acts as the symmetry group of the geometry. For background and motivation see the article on the Erlangen program. ==Formal definition== A Klein geometry is a pair where ''G'' is a Lie group and ''H'' is a closed Lie subgroup of ''G'' such that the (left) coset space ''G''/''H'' is connected. The group ''G'' is called the principal group of the geometry and ''G''/''H'' is called the space of the geometry (or, by an abuse of terminology, simply the ''Klein geometry''). The space of a Klein geometry is a smooth manifold of dimension :dim ''X'' = dim ''G'' − dim ''H''. There is a natural smooth left action of ''G'' on ''X'' given by : Clearly, this action is transitive (take ), so that one may then regard ''X'' as a homogeneous space for the action of ''G''. The stabilizer of the identity coset is precisely the group ''H''. Given any connected smooth manifold ''X'' and a smooth transitive action by a Lie group ''G'' on ''X'', we can construct an associated Klein geometry by fixing a basepoint ''x''0 in ''X'' and letting ''H'' be the stabilizer subgroup of ''x''0 in ''G''. The group ''H'' is necessarily a closed subgroup of ''G'' and ''X'' is naturally diffeomorphic to ''G''/''H''. Two Klein geometries and are geometrically isomorphic if there is a Lie group isomorphism so that . In particular, if ''φ'' is conjugation by an element , we see that and are isomorphic. The Klein geometry associated to a homogeneous space ''X'' is then unique up to isomorphism (i.e. it is independent of the chosen basepoint ''x''0). 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「Klein geometry」の詳細全文を読む スポンサード リンク
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