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 vanishing of the Cotton tensor for is necessary and sufficient condition for the manifold to be conformally flat, as with the Weyl tensor for . For the Cotton tensor is identically zero. The concept is named after Émile Cotton.
The proof of the classical result that for the vanishing of the Cotton tensor is equivalent to 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 .
Recently, the study of three-dimensional spaces is becoming of great interest, because the Cotton tensor restricts the relation between the Ricci tensor and the energy–momentum tensor of matter in the Einstein equations and plays an important role in the Hamiltonian formalism of general relativity.
Definition.
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 into a second order trace free tensor density
sometimes called the "Cotton–York tensor".
Properties.
Conformal rescaling.
Under conformal rescaling of the metric formula_3 for some scalar function formula_4. We see that the Christoffel symbols transform as
where formula_6 is the tensor
The Riemann curvature tensor transforms as
In formula_9-dimensional manifolds, we obtain the Ricci tensor by contracting the transformed Riemann tensor to see it transform as
Similarly the Ricci scalar transforms as
Combining all these facts together permits us to conclude the Cotton-York tensor transforms as
or using coordinate independent language as
where the gradient is plugged into the symmetric part of the Weyl tensor "W".
Symmetries.
The Cotton tensor has the following symmetries:
and therefore
In addition the Bianchi formula for the Weyl tensor can be rewritten as
where formula_17 is the positive divergence in the first component of "W".