Abstract
In the present paper, we initiate the study of \(*\)-\(\eta \)-Ricci soliton within the framework of Kenmotsu manifolds as a characterization of Einstein metrics. Here we display that a Kenmotsu metric as a \(*\)-\(\eta \)-Ricci soliton is Einstein metric if the soliton vector field is contact. Further, we have developed the characterization of the Kenmotsu manifold or the nature of the potential vector field when the manifold satisfies gradient almost \(*\)-\(\eta \)-Ricci soliton. Next, we deliberate \(*\)-\(\eta \)-Ricci soliton admitting \((\kappa ,\mu )^\prime \)-almost Kenmotsu manifold and proved that the manifold is Ricci flat and is locally isometric to \({\mathbb {H}}^{n+1}(-4)\times {\mathbb {R}}^n\). Finally we present some examples to decorate the existence of \(*\)-\(\eta \)-Ricci soliton, gradient almost \(*\)-\(\eta \)-Ricci soliton on Kenmotsu manifold.
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Acknowledgements
The second author was financially supported by UGC Senior Research Fellowship of India, Sr. No. 2061540940. Ref. No:21/06/2015(i)EU-V.
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Dey, S., Sarkar, S. & Bhattacharyya, A. \(*\)-\(\eta \)-Ricci soliton and contact geometry. Ricerche mat 73, 1203–1221 (2024). https://doi.org/10.1007/s11587-021-00667-0
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DOI: https://doi.org/10.1007/s11587-021-00667-0
Keywords
- Ricci flow
- \(\eta \)-Ricci soliton
- \(*\)-\(\eta \)-Ricci
- Gradient almost \(*\)-\(\eta \)-Ricci soliton
- Kenmotsu manifold
- \((\kappa , \mu )^\prime \)-almost Kenmotsu manifold