Critical point equation on four-dimensional compact manifolds

The aim of this article is to study the space of metrics with constant scalar curvature of volume 1 that satisfies the critical point equation Lg*(f)=Ric˚, for simplicity CPE metrics. It has been conjectured that every CPE metric must be Einstein. Here, we shall focus our attention for 4‐dimensional half conformally flat manifolds M4. In fact, we shall show that for a nontrivial f,M4 must be isometric to a sphere S4 and f is some height function on S4.