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In mathematics, in the field of geometry, a polar space of rank ''n'' (), or ''projective index'' , consists of a set ''P'', conventionally the set of points, together with certain subsets of ''P'', called ''subspaces'', that satisfy these axioms: * Every subspace is isomorphic to a projective geometry with and ''K'' a division ring. By definition, for each subspace the corresponding ''d'' is its dimension. * The intersection of two subspaces is always a subspace. * For each point ''p'' not in a subspace ''A'' of dimension of , there is a unique subspace ''B'' of dimension such that is -dimensional. The points in are exactly the points of ''A'' that are in a common subspace of dimension 1 with ''p''. * There are at least two disjoint subspaces of dimension . It is possible to define and study a slightly bigger class of objects using only relationship between points and lines: a polar space is a partial linear space (''P'',''L''), so that for each point ''p'' ∈ ''P'' and each line ''l'' ∈ ''L'', the set of points of ''l'' collinear to ''p'', is either a singleton or the whole ''l''. A polar space of rank two is a generalized quadrangle; in this case in the latter definition the set of points of a line ''l'' collinear to a point ''p'' is the whole ''l'' only if ''p ∈ l''. One recovers the former definition from the latter under assumptions that lines have more than 2 points, points lie on more than 2 lines, and there exist a line ''l'' and a point ''p'' not on ''l'' so that ''p'' is collinear to all points of ''l''. Finite polar spaces (where ''P'' is a finite set) are also studied as combinatorial objects. == Examples== * In a finite projective space over the field of size ''q'', with ''d'' odd and , the set of all points, with as subspaces the totally isotropic subspaces of an arbitrary symplectic polarity, forms a polar space of rank . * Let ''Q'' be a nonsingular quadric in with character ω. Then the index of ''Q'' will be . The set of all points on the quadric, together with the subspaces on the quadric, forms a polar space of rank . * Let ''H'' be a nonsingular Hermitian variety in . The index of ''H'' will be . The points on ''H'', together with the subspaces on it, form a polar space of rank . 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「polar space」の詳細全文を読む スポンサード リンク
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