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

TLDR: In this paper, the authors characterized the Einstein metrics in such broad classes of metrics as almost $$\eta $$¯¯ -Ricci solitons and almost $€  ¯¯¯¯ -RICci soliton on Kenmotsu manifolds, and generalized some known results.

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.