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In mathematics, a CR manifold is a differentiable manifold together with a geometric structure modeled on that of a real hypersurface in a complex vector space, or more generally modeled on an edge of a wedge. Formally, a CR manifold is a differentiable manifold ''M'' together with a preferred complex distribution ''L'', or in other words a subbundle of the complexified tangent bundle CT''M'' = T''M'' ⊗ C such that * (''L'' is formally integrable) * (L is almost Lagrangian). The bundle ''L'' is called a CR structure on the manifold ''M''. The abbreviation CR stands for Cauchy-Riemann or (Complex-Real ). ==Introduction and motivation== The notion of a CR structure attempts to describe ''intrinsically'' the property of being a hypersurface in complex space by studying the properties of holomorphic vector fields which are tangent to the hypersurface. Suppose for instance that ''M'' is the hypersurface of C2 given by the equation : where ''z'' and ''w'' are the usual complex coordinates on C2. The ''holomorphic tangent bundle'' of C2 consists of all linear combinations of the vectors : The distribution ''L'' on ''M'' consists of all combinations of these vectors which are ''tangent'' to ''M''. The tangent vectors must annihilate the defining equation for ''M'', so ''L'' consists of complex scalar multiples of : In particular, ''L'' consists of the holomorphic vector fields which annihilate ''F''. Note that ''L'' gives a CR structure on ''M'', for () = 0 (since ''L'' is one-dimensional) and since ∂/∂''z'' and ∂/∂''w'' are linearly independent of their complex conjugates. More generally, suppose that ''M'' is a real hypersurface in Cn, with defining equation ''F''(''z''1, ..., ''z''n) = 0. Then the CR structure ''L'' consists of those linear combinations of the basic holomorphic vectors on Cn: : which annihilate the defining function. In this case, for the same reason as before. Moreover, () ⊂ ''L'' since the commutator of holomorphic vector fields annihilating ''F'' is again a holomorphic vector field annihilating ''F''. 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「CR manifold」の詳細全文を読む スポンサード リンク
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