Cotton tensor

In 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 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 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

1. E Cotton, Sur les variétés à trois dimensions Ann. Fac. Sci. Toulouse II 1 385 1899
2. Luther Pfahler Eisenhart, Riemannian Geometry, Princeton Unversity Press, Princton, N.J. 1925, 1977, ISBN 0691080267
3. 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