On weakly Ricci symmetric lightlike hypersurfaces of indefinite Sasakian manifolds

Abstract

This paper deals with weakly Ricci-symmetric lightlike hypersurfaces of indefinite Sasakian manifolds, tangent to the structure vector field. We prove that, in a weakly Ricci symmetric lightlike $\eta$-Einstein (or Einstein) hypersurface of an indefinite Sasakian manifold, the associated 1-forms $\alpha$ and $\beta$ satisfy $\alpha +\beta =0$ (Theorem 4). Also, we show that there exist no weakly Ricci symmetric screen locally (or globally) conformal lightlike hypersurfaces of indefinite Sasakian manifolds with cyclic parallel Ricci tensor if $\alpha +\beta +\gamma$ is not everywhere zero (Theorem 5). A particular case of weakly Ricci symmetric condition is studied and we prove that a special weakly Ricci symmetric screen locally (or globally) conformal lightlike hypersurface cannot be $\eta$-Einstein (or Einstein) and under certain condition, it cannot be $(D\perp \langle \xi \rangle ,D\text{'})$-mixed-totally geodesic (Theorem 7).