Compact almost Co-Kähler manifolds and Ricci-Yamabe solitons

In this article we establish that if the metric g of a compact almost Co-K?hler manifold M2n+1 is a Ricci-Yamabe soliton whose potential vector field is point-wise collinear with the characteristic vector field, then M2n+1 is a K-almost Co-K?hler manifold under certain condition, whereas in dimension three the restriction is not required. It is proved that if a (2n+1)-dimensional (?, ?)-almost Co-K?hler manifold Mwith ? < 0 admits a Ricci-Yamabe soliton of gradient type, then M is a N(?)-almost Co-K?hler manifold. We also show the non-existence of gradient Ricci-Yamabe structures with D? = (??)? on a compact (?, ?)- almost Co-K?hler manifold with ? < 0. Then we establish that in a Co-K?hler 3-manifoldM3 with gradient Ricci-Yamabe solitons, the scalar curvature of the manifold is constant and also, either M3 is flat, or the gradient of the potential function is collinear with the characteristic vector field ?. Finally, we construct two non-trivial examples to ensure the existence of such solitons.