On the Geometry of Cotton Gravity

TL;DR

This paper explores the geometric structure of Cotton gravity on static spacetimes, introducing a generalized fluid model and analyzing conditions for geometric reduction and rigidity.

Contribution

It introduces the Cotton-$$-perfect fluid model, generalizes $$-static perfect fluid spaces, and studies geometric and curvature conditions in Cotton gravity.

Findings

Defined the Cotton-$$-perfect fluid structure.

Provided conditions for reduction to $$-SPFST.

Established rigidity results under curvature assumptions.

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