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 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 Aldersley.
Read more about Cotton Tensor: Definition
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