### Geometrical and Physical Properties of $W_2$-Symmetric and -Recurrent Manifolds

#### Abstract

The authors discusses mainly that the Riemannian

manifold $M^n$ admitting a unit preserving circle field $\xi$ in

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_{\alpha\beta\gamma\delta}=\frac{1}{d^{2}}[d_\beta d_\gamma

R_{\alpha\delta}-d_{\beta}d_{\delta}R_{\alpha\gamma}+d_{\alpha}d_{\delta}R_{\beta\gamma}-d_{\alpha}d_{\gamma}R_{\beta\delta}]

$.

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