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 n = 3 is necessary and sufficient condition for the manifold to be conformally flat, as with the Weyl tensor for n ≥ 4. For n < 3 the Cotton tensor is identically zero. The concept is named after Émile Cotton. 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 n = 3 is necessary and sufficient condition for the manifold to be conformally flat, as with the Weyl tensor for n ≥ 4. For n < 3 the Cotton tensor is identically zero. The concept is named after Émile Cotton. The proof of the classical result that for n = 3 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 (Aldersley 1979). 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. 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 into a second order trace free tensor density sometimes called the Cotton–York tensor. Under conformal rescaling of the metric g ~ = e 2 ω g {displaystyle { ilde {g}}=e^{2omega }g} for some scalar function ω {displaystyle omega } . We see that the Christoffel symbols transform as where S β γ α {displaystyle S_{eta gamma }^{alpha }} is the tensor The Riemann curvature tensor transforms as