On pseudo Ricci-symmetric manifolds (Q2770047)

The article is devoted to particular classes of pseudo Ricci-symmetric manifolds. A non-flat Riemannian manifold \((M^n,g)\) \((n>2)\) is called pseudo Ricci-symmetric if its Ricci tensor \(S\) is not identically zero and satisfies NEWLINE\[NEWLINE(\nabla_xS)(Y,Z)= 2\alpha(x) S(Y,Z)+ \alpha (Y)S(X,Z)+ \alpha(Z)S (Y,Z),NEWLINE\]NEWLINE where \(\alpha\) is a 1-form which is non-zero for every \(X,Y,Z\in \chi (M)\) and \(\nabla\) being the operator of covariant differentiation with respect to the metric \(g\).NEWLINENEWLINENEWLINEIt is shown that pseudo Ricci-symmetric manifolds satisfying \(\text{div} R=0\) (respectively \(\text{div} c=0)\) are Eistein (respectively Ricci flat) manifolds.