compact almost co-K\"ahler manifolds and Ricci-Yamabe solitons

Abstract

In this article we establish that if the metric g of a compact almost Co-K\"ahler manifold $\mathcal{M}^{2n+1}$ is a Ricci-Yamabe soliton whose potential vector field is point-wise collinear with the characteristic vector field, then $\mathcal{M}^{2n+1}$ is a K-almost Co-K\"ahler manifold under certain condition, whereas in dimension three the restriction is not required. It is prove that if a (2n+1)-dimensional $(\kappa,\mu)$-almost Co-K\"ahler manifold $\mathcal{M}$ with $\kappa <0$ permits a Ricci-Yamabe soliton of gradient type, then $\mathcal{M}$ is a $N(\kappa)$-almost Co-K\"ahler manifold. We also show the non-existence of gradient Ricci-Yamabe structures with $D\Psi=(\zeta \Psi)\zeta$ on a compact $(\kappa,\mu)$-almost Co-K\"ahler manifold with $\kappa<0$. Then we establish that in a co-K\"ahler 3-manifold $\mathcal{M}^{3}$ with gradient Ricci-Yamabe solitons, the scalar curvature of the manifold is constant and also, either $\mathcal{M}^{3}$ is flat, or the gradient of the potential function is collinear with $\zeta$. Finally, we construct two non-trivial examples to ensure the existence of such solitons.