Geometry of α-Cosymplectic Metric as ∗-Conformal η-Ricci–Yamabe Solitons Admitting Quarter-Symmetric Metric Connection

Abstract

The outline of this research article is to initiate the development of a ∗-conformal η-Ricci–Yamabe soliton in α-Cosymplectic manifolds according to the quarter-symmetric metric connection. Here, we have established some curvature properties of α-Cosymplectic manifolds in regard to the quarter-symmetric metric connection. Further, the attributes of the soliton when the manifold gratifies a quarter-symmetric metric connection have been displayed in this article. Later, we picked up the Laplace equation from ∗-conformal η-Ricci–Yamabe soliton equation when the potential vector field ξ of the soliton is of gradient type, admitting quarter-symmetric metric connection. Next, we evolved the nature of the soliton when the vector field’s conformal killing reveals a quarter-symmetric metric connection. We show an example of a 5-dimensional α-cosymplectic metric as a ∗-conformal η-Ricci–Yamabe soliton acknowledges quarter-symmetric metric connection to prove our results.

Geometry of $\ast$-$k$-Ricci-Yamabe soliton and gradient $\ast$-$k$-Ricci-Yamabe soliton on Kenmotsu manifolds

“…D. Dey [11] studied new type of soliton namely * -Ricci-Yamabe soliton on contact geometry. Recently, P. Zhang et al [56] have strained * -conformal η-Ricci-Yamabe soliton on α-cosymplectic manifolds. Siddiqi and Akyol [44] have discussed the notion of η-Ricci-Yamabe soliton to set up the geometrical structure on Riemannian submersions admitting η-Ricci-Yamabe soliton with the potential field.…”

Quarter-Symmetric Metric Connection on a Cosymplectic Manifold

“…The papers [14][15][16] are devoted to studying some special types of K-contact manifolds with respect to the quarter-symmetric metric connection. Among other things, the * -conformal η-Ricci-Yamabe soliton admitting a quartersymmetric metric connection on α-cosymplectic manifolds was studied in [17]. If α = 0, then the α-cosymplectic manifold reduces to the cosymplectic manifold.…”