Symmetry properties of lightlike hypersurfaces in indefinite Sasakian manifolds

Abstract

In this paper, we investigate symmetry properties of lightlike hypersurfaces in indefinite Sasakian manifolds, tangent to the structure vector field. Theorems on locally symmetric, semi-symmetric and Ricci semi-symmetric lightlike hypersurfaces are obtained. We show that, under some conditions, these hypersurfaces are totally geodesic. The non-existence conditions of specific lightlike hypersurfaces are given. We prove, under a certain condition, that in lightlike hypersurfaces of an indefinite Sasakian space form, tangent to the structure vector field, the local symmetry and semi-symmetry notions are equivalent. This equivalence is extended to the Ricci semi-symmetry notion when the lightlike hypersurfaces are considered to be $\eta$-totally umbilical.