The Lanczos potential and Chern-Simons theory

A new tensor $D$ is introduced which is constructed from the Lanczos potential and is of the same form as that of the Weyl tensor $C$ expressed in terms of the Lanczos potential except that covariant differentiation is replaced by transvection with a vector $v$. The new tensor has associated invariants $C\cdot D$ and $D^2$, the first of these can be interpreted as a Chern-Simons term for Weyl $C^2$ gravity. Both invariants allow various tensors to be constructed and some of the properties of these are investigated by using exact examples.