Certain almost Kenmotsu metrics satisfying the vacuum static equation

Abstract

We characterize the solutions of the vacuum static equation on a class of almost Kenmotsu manifolds. First, we prove that if the vacuum static equation has a non-trivial solution on (k, ?)?-almost Kenmotsu manifold, then it is locally isometric to some warped product spaces. Next, we prove that the vacuum static equation have only trivial solution on generalized(k, ?)-almost Kenmotsu manifold. At last, we consider the vacuum static equation on an almost Kenmotsu manifold with conformal Reeb foliation. We also provide some important examples of almost Kenmotsu manifolds that satisfies the vacuum static equation.