• Received : 2019.01.01
  • Accepted : 2019.10.16
  • Published : 2019.12.30


We define two types of null hypersurfaces as; isoparametric and quasi isoparametric null hypersurfaces of Lorentzian space forms, based on the two shape operators associated with a null hypersurface. We prove that; on any screen conformal isoparametric null hypersurface, the screen geodesics lie on circles in the ambient space. Furthermore, we prove that the screen distributions of isoparametric (or quasi isoparametric) null hypersurfaces with at most two principal curvatures are generally Riemannian products. Several examples are also given to illustrate the main concepts.


  1. L. J. Alias, S. C. de Almeida, and A. Brasil, Jr., Hypersurfaces with constant mean curvature and two principal curvatures in $S^{n+1}$, An. Acad. Brasil. Cienc. 76 (2004), no. 3, 489-497.
  2. C. Atindogbe, Scalar curvature on lightlike hypersurfaces, Balkan Society of Geometers, Geometry Balkan Press 2009, Applied Sciences, 11 (2009), 9-18.
  3. C. Atindogbe, J.-P. Ezin, and J. Tossa, Pseudoinversion of degenerate metrics, Int. J. Math. Math. Sci. 2003 (2003), no. 55, 3479-3501.
  4. C. Atindogbe, M. M. Harouna, and J. Tossa, Lightlike hypersurfaces in Lorentzian manifolds with constant screen principal curvatures, Afr. Diaspora J. Math. 16 (2014), no. 2, 31-45.
  5. J. Berndt, Real hypersurfaces with constant principal curvatures in complex hyperbolic space, J. Reine Angew. Math. 395 (1989), 132-141.
  6. E. Cartan, Familles de surfaces isoparametriques dans les espaces a courbure constante, Ann. Mat. Pura Appl. 17 (1938), no. 1, 177-191.
  7. T. E. Cecil and P. J. Ryan, Geometry of hypersurfaces, Springer Monographs in Mathematics, Springer, New York, 2015.
  8. B. Y. Chen, Riemannian submanifolds: A survey, arXiv:1307.1875[math.DG].
  9. G. de Rham, Sur la reductibilite d'un espace de Riemann, Comment. Math. Helv. 26 (1952), 328-344.
  10. J. Dong and X. Liu, Totally umbilical lightlike hypersurfaces in Robertson-Walker space-times, ISRN Geom. 2014 (2014), Art. ID 974695, 10 pp.
  11. K. L. Duggal and A. Bejancu, Lightlike submanifolds of semi-Riemannian manifolds and applications, Mathematics and its Applications, 364, Kluwer Academic Publishers Group, Dordrecht, 1996.
  12. K. L. Duggal and B. Sahin, Differential geometry of lightlike submanifolds, Frontiers in Mathematics, Birkhauser Verlag, Basel, 2010.
  13. J. Hahn, Isoparametric hypersurfaces in the pseudo-Riemannian space forms, Math. Z. 187 (1984), no. 2, 195-208.
  14. M. Hassirou, Kaehler lightlike submanifolds, J. Math. Sci. Adv. Appl. 10 (2011), no. 1-2, 1-21.
  15. D. H. Jin, Ascreen lightlike hypersurfaces of an indenite Sasakian manifold, J. Korean Soc. Math. Educ. Ser. B Pure Appl. Math. 20 (2013), no. 1, 25-35.
  16. M. Kimura and S. Maeda, Geometric meaning of isoparametric hypersurfaces in a real space form, Canad. Math. Bull. 43 (2000), no. 1, 74-78.
  17. D. N. Kupeli, Singular semi-Riemannian geometry, Mathematics and its Applications, 366, Kluwer Academic Publishers Group, Dordrecht, 1996.
  18. Z. Li and X. Xie, Space-like isoparametric hypersurfaces in Lorentzian space forms, Front. Math. China 1 (2006), no. 1, 130-137.
  19. M. A. Magid, Lorentzian isoparametric hypersurfaces, Pacic J. Math. 118 (1985), no. 1, 165-197.
  20. F. Massamba and S. Ssekajja, Quasi generalized CR-lightlike submanifolds of indenite nearly Sasakian manifolds, Arab. J. Math. (Springer) 5 (2016), no. 2, 87-101.
  21. M. Navarro, O. Palmas, and D. A. Solis, Null screen isoparametric hypersurfaces in Lorentzian space forms, Mediterr. J. Math. 15 (2018), no. 6, Art. 215, 14 pp.
  22. B. O'Neill, Semi-Riemannian Geometry, Pure and Applied Mathematics, 103, Academic Press, Inc., New York, 1983.
  23. K. Nomizu, Some results in E. Cartan's theory of isoparametric families of hypersurfaces, Bull. Amer. Math. Soc. 79 (1973), 1184-1188. 9904-1973-13371-3
  24. K. Nomizu, Elie Cartan's work on isoparametric families of hypersurfaces, in Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Part 1, Stanford Univ., Stanford, Calif., 1973), 191-200, Amer. Math. Soc., Providence, RI, 1975.