Almost $\eta$-Ricci solitons on Kenmotsu manifolds

Abstract

In this paper we characterize the Einstein metrics in such broader classes of metrics as almost $\eta$-Ricci solitons and $\eta$-Ricci solitons on Kenmotsu manifolds, and generalize some results of other authors. First, we prove that a Kenmotsu metric as an $\eta$-Ricci soliton is Einstein metric if either it is $\eta$-Einstein or the potential vector field $V$ is an infinitesimal contact transformation or $V$ is collinear to the Reeb vector field. Further, we prove that if a Kenmotsu manifold admits a gradient almost $\eta$-Ricci soliton with a Reeb vector field leaving the scalar curvature invariant, then it is an Einstein manifold. Finally, we present new examples of $\eta$-Ricci solitons and gradient $\eta$-Ricci solitons, which illustrate our results.