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http://dx.doi.org/10.4134/JKMS.j190001

GEOMETRY OF ISOPARAMETRIC NULL HYPERSURFACES OF LORENTZIAN MANIFOLDS  

Ssekajja, Samuel (School of Mathematics University of Witwatersrand)
Publication Information
Journal of the Korean Mathematical Society / v.57, no.1, 2020 , pp. 195-213 More about this Journal
Abstract
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.
Keywords
Null hypersurfaces; isoparametric hypersurfaces; geodesics;
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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. https://doi.org/10.1590/S0001-37652004000300003   DOI
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. https://doi.org/10.1155/S0161171203301309   DOI
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. http://projecteuclid.org/euclid.adjm/1413809901
5 J. Berndt, Real hypersurfaces with constant principal curvatures in complex hyperbolic space, J. Reine Angew. Math. 395 (1989), 132-141. https://doi.org/10.1515/crll.1989.395.132
6 E. Cartan, Familles de surfaces isoparametriques dans les espaces a courbure constante, Ann. Mat. Pura Appl. 17 (1938), no. 1, 177-191. https://doi.org/10.1007/BF02410700   DOI
7 T. E. Cecil and P. J. Ryan, Geometry of hypersurfaces, Springer Monographs in Mathematics, Springer, New York, 2015. https://doi.org/10.1007/978-1-4939-3246-7
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. https://doi.org/10.1007/BF02564308   DOI
10 J. Dong and X. Liu, Totally umbilical lightlike hypersurfaces in Robertson-Walker space-times, ISRN Geom. 2014 (2014), Art. ID 974695, 10 pp. https://doi.org/10.1155/2014/974695
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. https://doi.org/10.1007/978-94-017-2089-2
12 D. N. Kupeli, Singular semi-Riemannian geometry, Mathematics and its Applications, 366, Kluwer Academic Publishers Group, Dordrecht, 1996. https://doi.org/10.1007/978-94-015-8761-7
13 K. L. Duggal and B. Sahin, Differential geometry of lightlike submanifolds, Frontiers in Mathematics, Birkhauser Verlag, Basel, 2010. https://doi.org/10.1007/978-3-0346-0251-8
14 J. Hahn, Isoparametric hypersurfaces in the pseudo-Riemannian space forms, Math. Z. 187 (1984), no. 2, 195-208. https://doi.org/10.1007/BF01161704   DOI
15 M. Hassirou, Kaehler lightlike submanifolds, J. Math. Sci. Adv. Appl. 10 (2011), no. 1-2, 1-21.
16 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. https://doi.org/10.7468/jksmeb.2013.20.1.25
17 M. Kimura and S. Maeda, Geometric meaning of isoparametric hypersurfaces in a real space form, Canad. Math. Bull. 43 (2000), no. 1, 74-78. https://doi.org/10.4153/CMB-2000-011-3   DOI
18 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. https://doi.org/10.1007/s40065-016-0146-0   DOI
19 Z. Li and X. Xie, Space-like isoparametric hypersurfaces in Lorentzian space forms, Front. Math. China 1 (2006), no. 1, 130-137. https://doi.org/10.1007/s11464-005-0026-y   DOI
20 M. A. Magid, Lorentzian isoparametric hypersurfaces, Pacic J. Math. 118 (1985), no. 1, 165-197. http://projecteuclid.org/euclid.pjm/1102706671   DOI
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. https://doi.org/10.1007/s00009-018-1262-1   DOI
22 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.
23 B. O'Neill, Semi-Riemannian Geometry, Pure and Applied Mathematics, 103, Academic Press, Inc., New York, 1983.
24 K. Nomizu, Some results in E. Cartan's theory of isoparametric families of hypersurfaces, Bull. Amer. Math. Soc. 79 (1973), 1184-1188. https://doi.org/10.1090/S0002- 9904-1973-13371-3   DOI