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}]
$.