On the Geometry of Cotton Gravity

Abstract

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-$\ddot{I}†$-perfect fluid (C-$\ddot{I}†$-PF, for short) is a generalization to the regime of Cotton Gravity of the recently introduced notion of $\ddot{I}†$-static perfect fluid space-time ($\ddot{I}†$-SPFST). After discussing the variational origin of this system, we provide sufficient conditions for a C-$\ddot{I}†$-PF to reduce to a $\ddot{I}†$-SPFST. We also study the geometry of the level sets of the lapse function $f$ and we provide a rigidity result for C-$\ddot{I}†$-PFs under some curvature conditions. The role that Codazzi tensors hold in this theory is highlighted.

49 pages. Comments are welcome!