On nearly vacuum static equations in almost coKähler manifolds with applications to spacetimes

In the present article, we extend the notion of vacuum static equations on almost coKähler manifolds and rename them as nearly vacuum static equations. It is shown that if an \(\eta \) $\eta$ -Einstein almost coKähler manifold admits a non-trivial solution of a nearly vacuum static equation, then the solution must be a constant. In \((\kappa ,\mu )\) $(\kappa ,\mu )$ -almost coKähler manifolds, the non-trivial solutions of nearly vacuum static equations do not exist. We also apply nearly vacuum static equations on perfect fluid spacetimes as well as generalized Robertson–Walker spacetimes. Among others, it is shown that a perfect fluid spacetime admitting nearly vacuum static equations is of constant scalar curvature and a generalized Robertson–Walker spacetime obeying nearly vacuum static equations represents a dark matter era.