Closed vacuum static spaces with closed conformal vector fields

This article aims to classify closed vacuum static spaces with a non-Killing closed conformal vector field. We firstly provide several characterizations of the conditions under which the first derivative of the warping function fulfills the vacuum static equation. Then we establish an identity involving the characteristic function of a conformal vector field on a Riemannian manifold. As applications, we derive a rigidity theorem on closed Riemannian manifolds with a non-Killing closed conformal vector field under suitable conditions and classify closed vacuum static spaces admitting such a vector field.