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Cotton tensor
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<p>In <a href="/facts/Differential_geometry/c04XmyLu">differential geometry</a>, the Cotton tensor on a (pseudo)-<a href="/facts/Riemannian_manifold/5KYnmEgF">Riemannian manifold</a> of dimension <i>n</i> is a third-order <a href="/facts/Tensor_field/cAC6F6rF">tensor</a> concomitant of the <a href="/facts/Metric_tensor/tbqek1gJ">metric</a>. The vanishing of the Cotton tensor for <i>n</i> = 3 is <a href="/facts/Necessary_condition/G45aDCY9">necessary</a> and <a href="/facts/Sufficient_condition/G45aDCY9">sufficient condition</a> for the manifold to be locally <a href="/facts/Conformally_flat/QNAk4QA0">conformally flat</a>. By contrast, in dimensions <i>n</i> ≥ 4, the vanishing of the Cotton tensor is necessary but not sufficient for the metric to be conformally flat; instead, the corresponding necessary and sufficient condition in these higher dimensions is the vanishing of the <a href="/facts/Weyl_tensor/lrXNLWHC">Weyl tensor</a>, while the Cotton tensor just becomes a constant times the divergence of the Weyl tensor. For <i>n</i> < 3 the Cotton tensor is identically zero. The concept is named after <a href="/facts/%25C3%2589mile_Cotton/Ao2EYuMN">Émile Cotton</a>. </p><p>The proof of the classical result that for <i>n</i> = 3 the vanishing of the Cotton tensor is equivalent to the metric being conformally flat is given by <a href="/facts/Luther_P._Eisenhart/Iz3Ljcur">Eisenhart</a> using a standard <a href="/facts/Integrability_condition/PKp57zwz">integrability</a> 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 (Aldersley 1979). </p><p>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 <a href="/facts/Energy%25E2%2580%2593momentum_tensor/wWKrzjfI">energy–momentum tensor</a> of matter in the <a href="/facts/Einstein_equation/j2rJTzW9">Einstein equations</a> and plays an important role in the <a href="/facts/Hamiltonian_formalism/2ReerlNA">Hamiltonian formalism</a> of <a href="/facts/General_relativity/jVsoIg8X">general relativity</a>. </p>