On static manifolds and related critical spaces with zero radial Weyl curvature

Abstract

The aim of this paper is to study compact Riemannian manifolds $$(M,\,g)$$ that admit a non-constant solution to the system of equations $$\begin{aligned} -\Delta f\, g+Hess f-fRic=\mu Ric+\lambda g, \end{aligned}$$where Ric is the Ricci tensor of g whereas $$\mu $$ and $$\lambda $$ are two real parameters. More precisely, under assumption that $$(M,\,g)$$ has zero radial Weyl curvature, this means that the interior product of $$\nabla f$$ with the Weyl tensor W is zero, we shall provide the complete classification for the following structures: positive static triples, critical metrics of volume functional and critical metrics of the total scalar curvature functional.