Vacuum static spaces and Conformal vector fields

TL;DR

This paper proves that compact vacuum static spaces with non-trivial closed conformal vector fields are isometric to standard spheres, and provides criteria for conformal vector fields to be closed.

Contribution

It establishes a rigidity result linking conformal vector fields to the geometry of vacuum static spaces and extends to solutions of the critical point equation.

Findings

Vacuum static spaces with certain conformal vector fields are spherical.

Provides a criterion for conformal vector fields to be closed.

Extends results to solutions of the critical point equation.

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 ${\Bbb S}^n(c)$. 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.