Korean J. Math. Vol. 33 No. 4 (2025) pp.303-315
DOI: https://doi.org/10.11568/kjm.2025.33.4.303

$\ast$-Ricci-Yamabe Soliton and Contact Geometry

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Published: Dec 30, 2025
Keywords:
53D15, 35Q51, 53C25
Mathematics Subject Classification:
$N(k)$-contact manifold, Sasakian 3-manifold, $\ast$-Ricci-Yamabe soliton, Berger sphere, Positive-Sasakian, Infinitesimal automorphism

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Dibakar Dey
University of Calcutta

Abstract

It is well known that a unit sphere admits Sasakian 3-structure. Also, Sasakian manifolds are locally isometric to a unit sphere under several curvature and critical conditions. So, a natural question is: Does there exist any curvature or critical condition under which a Sasakian 3-manifold represents a geometrical object other than the unit sphere? In this regard, as an extension of the $\ast$-Ricci soliton, the notion of $\ast$-Ricci-Yamabe soliton is introduced and studied on two classes contact metric manifolds. A $(2n + 1)$-dimensional non-Sasakian $N(k)$-contact metric manifold admitting $\ast$-Ricci-Yamabe soliton is completely classified. Further, it is proved that if a Sasakian 3-manifold $M$ admits $\ast$-Ricci-Yamabe soliton $(g,V,\lambda,\alpha,\beta)$ under certain conditions on the soliton vector field $V$, then $M$ is $\ast$-Ricci flat, positive Sasakian and the transverse geometry of $M$ is Fano. In addition, the Sasakian 3-metric $g$ is homothetic to a Berger sphere and the soliton is steady. Also, the potential vector field $V$ is an infinitesimal automorphism of the contact metric structure.


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