A Type of Almost Co-Kahler Manifold Satisfying Vacuum Static Equation

Abstract

In this paper we consider $(\tilde{k}, \tilde{\mu})$ almost co-K\"ahler manifold satisfying the vacuum static equation. First we prove that if a $(\tilde{k}, \tilde{\mu})$ almost co-K\"ahler manifold satisfies the vacuum static equation then its scalar curvature satisfies certain relation or the solution of the equation is trivial. Next we prove that the value of the scalar curvature is constant considering the fact that the vacuum static equation has non-trivial solution.