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Cotton TensorIn differential geometry, the Cotton tensor on a (pseudo)-Riemannian manifold of dimension n is third-order tensor density concomitant of the metric. Like the Weyl tensor. Just as the vanishing of the Weyl tensor for n ≥ 4 is a necessary and sufficient condition for the manifold to be conformally flat, the same is true for the Cotton tensor for n = 3, while for n < 3 it is identically zero. In coordinates, and denoting the Ricci tensor by Rij 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 in to a second order trace free tensor field - .
sometimes called the Cotton-York tensor. The proof of 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. References - E Cotton, Sur les varits trois dimensions Ann. Fac. Sci. Toulouse II 1 385 1899
- Luther Pfahler Eisenhart, Riemannian Geometry, Princeton Unversity Press, Princton, N.J. 1925, 1977, ISBN 0691080267
- SJ Aldersley, Comments on certain divergence-free tensor densities in a 3-space, Journal of Mathematical Physics Vol 20(9) pp. 1905-1907. September 1979
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