Rigidity of Ricci solitons for $φ$-related metallic semi-Riemannian structures on para-Kenmotsu manifolds

Abstract

In this paper, we investigate Ricci solitons in the para-Kenmotsu setting with respect to a $φ$-related metallic semi-Riemannian metric arising from the underlying almost paracontact metric structure, under the assumption that the potential vector field is pointwise collinear with the Reeb vector field. We first establish the relation $\tS=S+2nβ\bigl(g-η\otimesη\bigr)$ between the Ricci tensors of the original metric $g$ and the $φ$-related metallic metric $\tg$. We then show that the Ricci soliton equation implies that the original metric $g$ is of $η$-Einstein type. By combining this condition with the contracted Bianchi identity, we obtain a rigidity result implying $f=0$, $λ=\frac{2n}{C}$, $S=-2ng$, and consequently $\tS=-\frac{2n}{C}\tg$. Thus, every Ricci soliton in this setting whose potential vector field is pointwise collinear with the Reeb vector field is necessarily trivial, and both $g$ and $\tg$ are Einstein. Finally, we provide two five-dimensional para-Kenmotsu examples: one realizing the resulting trivial Einstein Ricci soliton and another showing that failure of the required Einstein condition prevents the existence of such a soliton.