Weakly Einstein and weakly $η$-Einstein structures on almost contact metric three-manifolds

Abstract

We study weakly Einstein and weakly $η$-Einstein structures on almost contact metric three-manifolds. We first obtain a pointwise structural description of the Ricci operator $Q$ on a weakly $η$-Einstein almost contact metric three-manifold. More precisely, at each point, either $Q$ has the $η$-Einstein form, $Qξ=0$, or the metric is weakly Einstein. We then prove that if a contact metric three-manifold satisfies $Qξ=0$, then either $Q=0$ or $\operatorname{rank}Q=1$. Hence, such a metric is either flat or weakly Einstein but not Einstein. We also construct examples showing that the contact metric assumption is essential: there exist almost contact metric three-manifolds satisfying $Qξ=0$ whose Ricci operators have rank two. Finally, we classify simply connected homogeneous contact metric three-manifolds that are weakly Einstein.