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.
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Patra, D.S., Rovenski, V. Almost \(\eta \)-Ricci solitons on Kenmotsu manifolds. European Journal of Mathematics 7, 1753–1766 (2021). https://doi.org/10.1007/s40879-021-00474-9
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DOI: https://doi.org/10.1007/s40879-021-00474-9
Keywords
- Almost contact structure
- Einstein manifold
- \(\eta \)-Ricci soliton
- Infinitesimal contact transformation