On pseudo Ricci symmetric manifolds

Let (Mn, g) be a Riemannian manifold with metric g. If its Ricci tensor S satisfies the equation ∇xS(Y, Z)=2A(X)S(Y, Z)+A(Y)S(X, Z)+A(Z)S(Y, X), where A is a non-zero 1-form, g(x, P) for every vector field X and ∇ is the covariant differentiation, it will be called an n-dimensional pseudo Ricci symmetric manifold (PRS)n. It is shown that the existence of such (PRS)n and several properties of (PRS)n are established, e.g. that if the scalar curvature r is a constant, then it must be equal to 0 or otherwise if r≠0 then its associated 1-form A must be closed. 4 refs