Bulletin of the
Korean Mathematical Society BKMS

In this paper, we study some geometric structures of a weakly cyclic $Z$-symmetric manifold (briefly, $[WCZS]_{n}$). More precisely, we prove that a conformally flat $[WCZS]_{n}$ satisfying certain conditions is special conformally flat and hence the manifold can be isometrically immersed in an Euclidean manifold $E^{n+1}$ as a hypersurface if the manifold is simply connected. Also we show that there exists a $[WCZS]_{4}$ with one parameter family of its associated 1-forms.

Keywords: weakly cyclic $Z$-symmetric manifolds, quasi Einstein, conformal Killing vector field, parallel vector field, special conformally flat, hypersurface

MSC numbers: 53A55, 53B20