Cotton tensor

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 concept is named after Émile Cotton. 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

$C_{ijk} = \nabla_{k} R_{ij} - \nabla_{j} R_{ik} + \frac{1}{2(n-1)}\left( \nabla_{j}Rg_{ik} - \nabla_{k}Rg_{ij}\right).$

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 density

$C_i^j = \nabla_{k} \left( R_{li} - \frac{1}{4} Rg_{li}\right)\epsilon^{klj},$

sometimes called the Cotton–York tensor.

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.

Properties

Conformal rescaling

Under conformal rescaling of the metric $\tilde{g} = e^{2\omega} g$ for some scalar function ω the Cotton-York tensor transforms as

$\tilde{C} = C \; - \; \operatorname{grad} \, \omega \; \lrcorner \; W,$^{[citation needed]}

where the gradient is plugged into the symmetric part of the Weyl tensor W.

Symmetries

The Cotton tensor has the following symmetries:

$C_{ijk} = - C_{ikj} \,$

and therefore

$C_{[ijk]} = 0. \,$

In addition the Bianchi formula for the Weyl tensor for can be rewritten as

$\delta W = (3-n) C, \,$

where δ is the positive divergence in the first component of W.

References

• Aldersley, S. J. (1979). "Comments on certain divergence-free tensor densities in a 3-space". Journal of Mathematical Physics 20 (9): 1905–1907. Bibcode 1979JMP....20.1905A. doi:10.1063/1.524289. 
• Cotton, É. (1899). "Sur les variétés à trois dimensions". Ann. Fac. Sci. Toulouse. II 1 (4): 385–438. http://www.numdam.org/numdam-bin/fitem?id=AFST_1899_2_1_4_385_0. 
• Eisenhart, Luther P. (1977) [1925]. Riemannian Geometry. Princeton, NJ: Princeton University Press. ISBN 0691080267.