On weakly Einstein almost contact manifolds

Abstract

In this article we study almost contact manifolds admitting weakly Einstein metrics. We first prove that if a (2n+1)-dimensional Sasakian manifold admits a weakly Einstein metric then its scalar curvature $s$ satisfies $-6⩽s⩽6$ for $n=1$ and $-2n(2n+1)\frac{4n^2-4n+3}{4n^2-4n-1}⩽s⩽2n(2n+1)$ for $n⩾2$. Secondly, for a (2n+1)-dimensional weakly Einstein contact metric $(\kappa ,\mu )$-manifold with $\kappa <1$, we prove that it is flat or is locally isomorphic to the Lie group $SU(2)$, $SL(2)$, or $E(1,1)$ for $n=1$ and that for $n⩾2$ there are no weakly Einstein metrics on contact metric $(\kappa ,\mu )$-manifolds with $0<\kappa <1$. For $\kappa <0$, we get a classification of weakly Einstein contact metric $(\kappa ,\mu )$-manifolds. Finally, it is proved that a weakly Einstein almost cosymplectic $(\kappa ,\mu )$-manifold with $\kappa <0$ is locally isomorphic to a solvable non-nilpotent Lie group.

13 pages, accepted by Journal of the Korean Mathematical Society