Vacuum static spaces and Conformal vector fields

Abstract

In this paper, we show that if a compact $n$-dimensional vacuum static space $(M^n,g,f)$ admits a non-trivial closed conformal vector field $V$, then $(M,g)$ is isometric to a standard sphere ${\mathbb{S}}^n(c)$. We also prove that if a pair $(g,f)$ of a Riemannian metric and a function defined on a compact $n$-dimensional manifold $M^n$ satisfies the critical point equation and $(M,g)$ admits a non-trivial closed conformal vector field $V$, we have the same result. Finally, we prove a criterion for a nontrivial conformal vector field to be closed.

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