Geometrical and physical properties of W2-symmetric and -recurrent manifolds

The authors discuss mainly that the Riemannian manifold Mn admitting a unit preserving circle field ? in present paper. A sufficient and necessary condition is given an Einstein by imposing some conditions on W2 curvature tensor. Further, this paper obtains algebra representation of tensors W2-recurrent R???? = 1/d2 [d?d?R?? d?d?R?? + d?d?R??].