Geometry of warped product lightlike submanifolds of indefinite nearly Kaehler manifolds

Abstract

The aim of present paper is to study geometry of warped product SCR-lightlike submanifolds of indefinite nearly Kaehler manifolds. We prove the non-existence of warped product SCR-lightlike submanifolds of the type $$N_{\bot } \times _{f} N_{T}$$ in an indefinite nearly Kaehler manifold. We find a necessary and sufficient condition for a SCR-lightlike submanifold of an indefinite nearly Kaehler manifold to be a SCR-lightlike warped product submanifold of the type $$N_{T} \times _{f} N_{\bot }$$ . We give some characterizations in terms of the canonical structures T and $$\omega $$ on a SCR-lightlike submanifold of an indefinite nearly Kaehler manifold under which it reduces to a SCR-lightlike warped product submanifold. We also prove that for a proper SCR-lightlike warped product submanifold of an indefinite nearly Kaehler manifold, the induced connection $$\nabla $$ can never be a metric connection. Finally, we obtain a sharp inequality for the squared norm of the second fundamental form in terms of the warping function for a SCR-lightlike warped product submanifold of an indefinite nearly Kaehler manifold.