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In mathematics, the Fubini–Study metric is a Kähler metric on projective Hilbert space, that is, complex projective space CP''n'' endowed with a Hermitian form. This metric was originally described in 1904 and 1905 by Guido Fubini and Eduard Study. A Hermitian form in (the vector space) C''n''+1 defines a unitary subgroup U(''n''+1) in GL(''n''+1,C). A Fubini–Study metric is determined up to homothety (overall scaling) by invariance under such a U(''n''+1) action; thus it is homogeneous. Equipped with a Fubini–Study metric, CP''n'' is a symmetric space. The particular normalization on the metric depends on the application. In Riemannian geometry, one uses a normalization so that the Fubini–Study metric simply relates to the standard metric on the (2''n''+1)-sphere. In algebraic geometry, one uses a normalization making CP''n'' a Hodge manifold. ==Construction== The Fubini–Study metric arises naturally in the quotient space construction of complex projective space. Specifically, one may define CP''n'' to be the space consisting of all complex lines in C''n''+1, i.e., the quotient of C''n''+1\ by the equivalence relation relating all complex multiples of each point together. This agrees with the quotient by the diagonal group action of the multiplicative group C * = C \ : : This quotient realizes C''n''+1\ as a complex line bundle over the base space CP''n''. (In fact this is the so-called tautological bundle over CP''n''.) A point of CP''n'' is thus identified with an equivalence class of (''n''+1)-tuples () modulo nonzero complex rescaling; the ''Z''''i'' are called homogeneous coordinates of the point. Furthermore, one may realize this quotient in two steps: since multiplication by a nonzero complex scalar ''z'' = ''R'' ''e''iθ can be uniquely thought of as the composition of a dilation by the modulus ''R'' followed by a counterclockwise rotation about the origin by an angle , the quotient C''n''+1 → CP''n'' splits into two pieces. : where step (a) is a quotient by the dilation Z ~ ''R''Z for ''R'' ∈ R+, the multiplicative group of positive real numbers, and step (b) is a quotient by the rotations Z ~ ''e''iθZ. The result of the quotient in (a) is the real hypersphere ''S''2''n''+1 defined by the equation |Z|2 = |''Z''0|2 + ... + |''Z''''n''|2 = 1. The quotient in (b) realizes CP''n'' = ''S''2''n''+1/''S''1, where ''S''1 represents the group of rotations. This quotient is realized explicitly by the famous Hopf fibration ''S''1 → ''S''2''n''+1 → CP''n'', the fibers of which are among the great circles of . 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「Fubini–Study metric」の詳細全文を読む スポンサード リンク
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