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In differential geometry, representation theory and harmonic analysis, a symmetric space is a smooth manifold whose group of symmetries contains an inversion symmetry about every point. There are two ways to formulate the inversion symmetry: via Riemannian geometry or via Lie theory. The Lie-theoretic definition is more general and more algebraic. In Riemannian geometry, the inversions are geodesic symmetries, and these are required to be isometries, leading to the notion of a Riemannian symmetric space. More generally, in Lie theory a symmetric space is a homogeneous space ''G''/''H'' for a Lie group ''G'' such that the stabilizer ''H'' of a point is an open subgroup of the fixed point set of an involution of ''G''. This definition includes (globally) Riemannian symmetric spaces and pseudo-Riemannian symmetric spaces as special cases. Riemannian symmetric spaces arise in a wide variety of situations in both mathematics and physics. They were first studied extensively and classified by Élie Cartan. More generally, classifications of irreducible and semisimple symmetric spaces have been given by Marcel Berger. They are important in representation theory and harmonic analysis as well as differential geometry. ==Definition using geodesic symmetries== Let ''M'' be a connected Riemannian manifold and ''p'' a point of ''M''. A map ''f'' defined on a neighborhood of ''p'' is said to be a geodesic symmetry, if it fixes the point ''p'' and reverses geodesics through that point, i.e. if ''γ'' is a geodesic and then It follows that the derivative of the map at ''p'' is minus the identity map on the tangent space of ''p''. On a general Riemannian manifold, ''f'' need not be isometric, nor can it be extended, in general, from a neighbourhood of ''p'' to all of ''M''. ''M'' is said to be locally Riemannian symmetric if its geodesic symmetries are in fact isometric, and (globally) Riemannian symmetric if in addition its geodesic symmetries are defined on all of ''M''. 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「Symmetric space」の詳細全文を読む スポンサード リンク
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