Recurrent Z forms on riemannian and Kaehler Manifolds

In this paper, we introduce a new kind of Riemannian manifold that generalize the concept of weakly Z-symmetric and pseudo-Z-symmetric manifolds. First a Z form associated to the Z tensor is defined. Then the notion of Z recurrent form is introduced. We take into consideration Riemannian manifolds in which the Z form is recurrent. This kind of manifold is named (ZRF)(n). The main result of the paper is that the closedness property of the associated covector is achieved also for rank(Z(kl)) > 2. Thus the existence of a proper concircular vector in the conformally harmonic case and the form of the Ricci tensor are confirmed for(ZRF)(n) manifolds with rank(Z(kl)) > 2. This includes and enlarges the corresponding results already proven for pseudo-Z-symmetric (PZS)(n) and weakly Z-symmetric manifolds (WZS)(n) in the case of non-singular Z tensor. In the last sections we study special conformally flat (ZRF)(n) and give a brief account of Z recurrent forms on Kaehler manifolds.