A remark on the axiom of spheres with lightlike submanifolds and application of Jacobi equation

In this paper we improve the axiom of r-planes(r-spheres) for lightlike submanifolds given by Kumar, Rani and Nagaich. Moreover we introduce another axiom called axiom of r-submanifolds. We prove that if a lightlike submanifold (M,g,S(TM),S(TM?)) of a semi-Riemannian manifold (M,g) satisfies axiom of r-submanifolds, then it is a real space form. We also use Jacobi equation and obtain some related results to totally umbilical lightlike submanifold of semi-Riemannian manifold.