Cotton tensor

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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. 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

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.

[edit] References

  1. E Cotton, Sur les variétés à trois dimensions Ann. Fac. Sci. Toulouse II 1 385 1899 online version
  2. Luther Pfahler Eisenhart, Riemannian Geometry, Princeton University Press, Princeton, N.J. 1925, 1977, ISBN 0-691-08026-7
  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
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