Volume 37 • Number 2 • 2014   • A Remark on the Axiom of Spheres with Lightlike Submanifolds and Application of Jacobi Equation Mohammed Jamali, Jae-Bok Jun, Mohammad Hasan Shahid and Mobin Ahmad Abstract. 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 $(M,g,S(TM),S(TM^{\perp}))$ of a semi-Riemannian manifold $(\bar{M},\bar{g})$ 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. 2010 Mathematics Subject Classification: 53B25, 53C40, 53C50 Full text: PDF