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In differential geometry, the Cotton tensor on a (pseudo)-Riemannian manifold of dimension ''n'' is a third-order tensor concomitant of the metric, like the Weyl tensor. The vanishing of the Cotton tensor for ''n''=3 is necessary and sufficient condition for the manifold to be conformally flat, as with the Weyl tensor for ''n''≥4. For ''n''<3 the Cotton tensor is identically zero. The concept is named after Émile Cotton. The proof of the classical result that for ''n'' = 3 the vanishing of the Cotton tensor is equivalent to the metric being conformally flat is given by Eisenhart using a standard integrability argument. This tensor density is uniquely characterized by its conformal properties coupled with the demand that it be differentiable for arbitrary metrics, as shown by . Recently, the study of three-dimensional spaces is becoming of great interest, because the Cotton tensor restricts the relation between the Ricci tensor and the energy-momentum tensor of matter in the Einstein equations and plays an important role in the Hamiltonian formalism of general relativity. == Definition == In coordinates, and denoting the Ricci tensor by ''R''''ij'' and the scalar curvature by ''R'', the components of the Cotton tensor are : The Cotton tensor can be regarded as a vector valued 2-form, and for ''n'' = 3 one can use the Hodge star operator to convert this into a second order trace free tensor density : sometimes called the ''Cotton–York tensor''. 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「Cotton tensor」の詳細全文を読む スポンサード リンク
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