Ricci-Yamabe solitons on the Lie group $\Sol \times \mathbb{R}^n$

Abstract

In this article, we study Ricci-Yamabe solitons on the Lie group $\mathrm{Sol}\times {\mathbb{R}}^n$ equipped with a natural left-invariant Riemannian metric, explicitly determining the vector fields that characterize them. We then deduce that, in the case of a Ricci soliton, it is expanding, whereas in the case of a Yamabe soliton, it is shrinking. Finally, we show that if this Lie group is a gradient Ricci-Yamabe soliton, the vector field belongs to $\mathrm{Span}\{{\partial }_{t_1},\dots ,{\partial }_{t_n}\}$, and we explicitly provide the Perelman potential.