Almost $$\eta $$-Ricci solitons on Kenmotsu manifolds

Abstract

We characterize the Einstein metrics in such broad classes of metrics as almost \(\eta \)-Ricci solitons and \(\eta \)-Ricci solitons on Kenmotsu manifolds, and generalize some known results. 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.