∗-Ricci Tensor onα-Cosymplectic Manifolds

Abstract

In this paper, we study α-cosymplectic manifold$M$admitting$\ast$-Ricci tensor. First, it is shown that a$\ast$-Ricci semisymmetric manifold$M$is$\ast$-Ricci flat and a$\varphi$-conformally flat manifold$M$is an$\eta$-Einstein manifold. Furthermore, the$\ast$-Weyl curvature tensor${\mathcal{W}}^{\ast }$on$M$has been considered. Particularly, we show that a manifold$M$with vanishing$\ast$-Weyl curvature tensor is a weak$\varphi$-Einstein and a manifold$M$fulfilling the condition$R\left(E_1,E_2\right)\cdot {\mathcal{W}}^{\ast }=0$is$\eta$-Einstein manifold. Finally, we give a characterization for α-cosymplectic manifold$M$admitting$\ast$-Ricci soliton given as to be nearly quasi-Einstein. Also, some consequences for three-dimensional cosymplectic manifolds admitting$\ast$-Ricci soliton and almost$\ast$-Ricci soliton are drawn.