Let R_{hl} be the Ricci tensor for the metric g.  The Cotton tensor is defined in components by C^{ij} = e^{ihk} g^{jl} nabla_k R_{hl} (symmetrize on i, j).  Here e^{ihk}  denotes the contravariant permutation symbol and nabla_k R_{hl} is the covariant derivative of the Ricci tensor with respect to the Christoffel connection.

The Cotton tensor is symmetric, trace-free, divergence-free and a relative conformally invariant of the metric.

This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form CottonTensor(...) only after executing the command with(DifferentialGeometry) and with(Tensor) in that order.  It can always be used in the long form DifferentialGeometry:-Tensor:-CottonTensor.