Geodesic
A Remark on the Axiom of Spheres with Lightlike Submanifolds and Application of Jacobi Equation
Bulletin of the Malaysian Mathematical Society, Tome 37 (2014) no. 2 Cet article a éte moissonné depuis la source Bulletin of the Malaysian Mathematical Society website

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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.
Classification : 53B25, 53C40, 53C50 
@article{BMMS_2014_37_2_a22,
     author = {Mohammed Jamali and Jae-Bok Jun and Mohammad Hasan Shahid and Mobin Ahmad},
     title = {A {Remark} on the {Axiom} of {Spheres} with {Lightlike} {Submanifolds} and {Application} of {Jacobi} {Equation}},
     journal = {Bulletin of the Malaysian Mathematical Society},
     year = {2014},
     volume = {37},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/BMMS_2014_37_2_a22/}
}
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Mohammed Jamali; Jae-Bok Jun; Mohammad Hasan Shahid; Mobin Ahmad. A Remark on the Axiom of Spheres with Lightlike Submanifolds and Application of Jacobi Equation. Bulletin of the Malaysian Mathematical Society, Tome 37 (2014) no. 2. http://geodesic.mathdoc.fr/item/BMMS_2014_37_2_a22/