Characterization of a Sasakian manifold admitting nearly vacuum static equations

In this study, we explore the geometric structure of nearly vacuum static equations (NVSE) within the context of Sasakian geometry. Focusing on η-Einstein Sasakian manifolds, we demonstrate that the presence of such equations implies constant scalar curvature, with solutions either being trivial or reducing the manifold to an Einstein one. Furthermore, we investigate the implications in dimension three, showing that non-trivial solutions of nearly vacuum static equations necessarily transform the Sasakian manifold into a Sasakian space form. Notably, when such a manifold admits a non-isometric conformal vector field, it is shown to possess constant sectional curvature equal to one. These results contribute to a deeper understanding of the interplay between curvature conditions and static equations in contact metric geometry. Furthermore, we rigorously prove several theorems characterizing generalized quasi-Einstein Sasakian manifolds admitting nearly vacuum static equations. Finally, to substantiate the obtained results, we conclude with an explicit example of a 3-dimensional Sasakian manifold that satisfies the nearly vacuum static equation, thereby illustrating the geometric constraints and confirming our theoretical findings.