Ricci solitons and certain related metrics on almost co-Kaehler manifolds

In the paper, we study a Ricci soliton and a generalized $m$-quasi-Einstein metric on almost co-Kaehler manifold $M$ satisfying a nullity condition. First, we consider a non-co-Kaehler $(\kappa, \mu)$-almost co-Kaehler metric as a Ricci soliton and prove that the soliton is expanding with $\lambda=-2n\kappa$ and the soliton vector field $X$ leaves the structure tensors $\eta,\xi$ and $\varphi$ invariant. This result extends Theorem 5.1 of [32]. We construct an example to show the existence of a Ricci soliton on $M$. Finally, we prove that if $M$ is a generalized $(\kappa, \mu)$-almost co-Kaehler manifold of dimension higher than 3 such that $h\neq 0$, then the metric of $M$ can not be a generalized $m$-quasi-Einstein metric, and this recovers the recent result of Wang [37, Theorem 4.1] as a special case.