On Ricci tensor of weakly concircular symmetric Riemannian spaces

In the first section of this paper, the definitions of weakly symmetric and weakly concircular symmetric Riemannian space are given. In the second section, we will give an example to prove the existence of weakly symmetric spaces. In the third part of it, we prove that some theorems related to the Ricci associate vector of a weakly concircular symmetric space and 1-forms of this manifold. In the fourth part of this paper, we prove a theorem about a weakly concircular symmetric Riemannian space with a recurrent vector field. In the last part of this paper, we prove that if a weakly symmetric space of positive definite metric has cyclic parallel Ricci tensor, then this space is an Einstein space for n < 3.