Comments on certain divergence-free tensor densities in a 3-space

Abstract

It is well known that a necessary and sufficient condition for the conformal flatness of a three-dimensional pseudo-Riemannian manifold can be expressed in terms of the vanishing of a third-order tensor density concomitant of the metric which has contravariant valence 2. This was first discovered by Cotton in 1899. It is shown that Cotton's tensor density is not the Euler-Lagrange expression corresponding to a scalar density built from one metric tensor. This tensor density is shown to be uniquely characterized by its conformal properties coupled with the demand that it be differentiable for arbitrary metrics. © 1979 American Institute of Physics.