Closed vacuum static spaces with closed conformal vector fields

TL;DR

This paper classifies closed vacuum static spaces that admit non-Killing closed conformal vector fields, providing new characterizations and a rigidity theorem to deepen understanding of their geometric structure.

Contribution

It introduces new characterizations of the warping function's derivatives and classifies such spaces, advancing the understanding of their geometric and conformal properties.

Findings

Rigidity theorem for closed Riemannian manifolds with non-Killing closed conformal vector fields

Classification of closed vacuum static spaces with such vector fields

Derived identities involving the characteristic function of conformal vector fields

Abstract

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.