Cotton tensor

Summary

The Cotton tensor is defined on a pseudo-Riemannian manifold of dimension at least $3$. It is a $3$-index tensor built from the covariant derivatives of the Schouten tensor, a modified Ricci curvature. In dimension $3$ the Weyl tensor is always zero, and the Cotton tensor takes over its role. A $3$-dimensional metric is locally conformally flat if and only if its Cotton tensor vanishes. In higher dimensions it is, up to a constant factor, the divergence of the Weyl tensor.