Almost $η$-Ricci solitons on Kenmotsu manifolds

Abstract

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

Geometry of almost contact metrics as almost $*$-Ricci solitons

“…First, Ghosh [13] proved that a Kenmotsu metric as a Ricci soliton is Einstein if the metric is η-Einstein [11] or the potential vector field V is contact. Recently, Patra-Rovenski [22] proved that a Kenmotsu metric as an η-Ricci soliton is Einstein if either it is η-Einstein or the potential vector field V is an infinitesimal contact transformation or V is collinear to the Reeb vector field.…”