On the Geometry of Cotton Gravity

We analyze the geometry of the field equations of Cotton gravity (for a quite general energy-momentum tensor) on a static space-time. In particular, we describe the local structure of the spatial Riemannian factor. This structure, that we call Cotton- \(\varphi \) $\phi$ -perfect fluid (C- \(\varphi \) $\phi$ -PF, for shorts) is a generalization to the regime of Cotton Gravity of the recently introduced notion of \(\varphi \) $\phi$ -static perfect fluid space-time ( \(\varphi \) $\phi$ -SPFST). After discussing the variational origin of this system, we provide sufficient conditions for a C- \(\varphi \) $\phi$ -PF to reduce to a \(\varphi \) $\phi$ -SPFST. We also study the geometry of the level sets of the lapse function f and we provide a rigidity result for C- \(\varphi \) $\phi$ -PFs under some curvature conditions. The role that Codazzi tensors hold in this theory is highlighted.