J. Korean Math. Soc. Published online October 15, 2019

Xiaomin Chen
China Uniersity of Petroleum-Beijing

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 $-6\leq s \leq 6$ for $n=1$ and $-2n(2n+1)\frac{4n^2-4n+3}{4n^2-4n-1}\leq s \leq 2n(2n+1)$ for $n\geq2$. 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 $n\geq2$ there are no weakly Einstein metrics on contact metric $(\kappa,\mu)$-manifolds with $\kappa<1$. 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.

Keywords : weakly Einstein metric; Sasakian manifold; $(\kappa,\mu)$-manifold; almost cosymplectic manifold; Einstein manifold.