The authors discuss mainly that the Riemannian manifold M n admitting a unit preserving circle field ξ in the present paper. A sufficient and necessary condition is given that Riemannian manifold M n is an Einstein manifold by imposing some conditions on W 2 curvature tensor. Further, this paper obtains the algebra representation of curvature tensors of a W 2-recurrent Riemannian manifold M n given by R αβγδ = 1 d 2 [d β d γ R αδ − d β d δ R αγ + d α d δ R βγ − d α d γ R βδ ]