Geometry of isoparametric null hypersurfaces of Lorentzian manifolds
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We define two types of null hypersurfaces as; {\it isoparametric} and {\it 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.
Abstract
We define two types of null hypersurfaces as; {\it isoparametric} and {\it 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.
참고문헌 (24)
- L. J. Alas, "Hypersurfaces with constant mean curvature and two principal curvatures in Sn+1", An. Acad. Brasil. Cienc., 76(3), pp.489~497 (2004)[DOI]
- C. Atindogbe, "Scalar curvature on lightlike hypersurfaces", Applied Sciences, 11, pp.9~18 (2009), Geometry Balkan Press
- C. Atindogbe, "Pseudoinversion of degenerate metrics", Int. J. Math. Math. Sci, 2003(55), pp.3479~3501 (2003)[DOI]
- C. Atindogbe, "Lightlike hypersurfaces in Lorentzian manifolds with constant screen principal curvatures", Afr. Diaspora J. Math, 16(2), pp.31~45 (2014)
- J. Berndt, "Real hypersurfaces with constant principal curvatures in complex hyperbolic space", J. Reine Angew. Math, 395, pp.132~141 (1989)[DOI]
- E. Cartan, "Familles de surfaces isoparametriques dans les espaces a courbure constante", Ann. Mat. Pura Appl, 17(1), pp.177~191 (1938)[DOI]
- T. E. Cecil, "Geometry of hypersurfaces" (2015), Springer
- , ~
- G. de Rham, "Sur la reductibilite d'un espace de Riemann", Comment. Math. Helv, 26, pp.328~344 (1952)[DOI]
- J. Dong, "Totally umbilical lightlike hypersurfaces in Robertson-Walker space-times", ISRN Geom, 2014, pp.10~ (2014)[DOI]
- K. L. Duggal, "Mathematics and its Applications, 364" (1996), Kluwer Academic Publishers Group
- K. L. Duggal, "Dierential geometry of lightlike submanifolds" (2010), Frontiers in Mathematics, Birkhauser Verlag
- J. Hahn, "Isoparametric hypersurfaces in the pseudo-Riemannian space forms", Math. Z, 187(2), pp.195~208 (1984)[DOI]
- M. Hassirou, "Kaehler lightlike submanifolds", J. Math. Sci. Adv. Appl, 10(1-2), pp.1~21 (2011)
- 진대호, "ASCREEN LIGHTLIKE HYPERSURFACES OF AN INDEFINITE SASAKIAN MANIFOLD", 순수 및 응용수학, 20(1), pp.25~35 (2013), 한국수학교육학회
- Makoto Kimura, "Geometric Meaning of Isoparametric Hypersurfaces in a Real Space Form", Canadian Mathematical Bulletin, 43(1), pp.74~78 (2000), Canadian Mathematical Society[DOI]
- D. N. Kupeli, "Singular semi-Riemannian geometry" (1996), Kluwer Academic Publishers Group
- Z. Li, "Space-like isoparametric hypersurfaces in Lorentzian space forms", Front. Math. China, 1(1), pp.130~137 (2006)[DOI]
- M. A. Magid, "Lorentzian isoparametric hypersurfaces", Pacic J. Math, 118(1), pp.165~197 (1985)
- F. Massamba, "Quasi generalized CR-lightlike submanifolds of indenite nearly Sasakian manifolds", Arab. J. Math. (Springer), 5(2), pp.87~101 (2016)[DOI]
- M. Navarro, "Null screen isoparametric hypersurfaces in Lorentzian space forms", Mediterr. J. Math, 15(6), pp.14~ (2018)[DOI]
- B. O'Neill, "Pure and Applied Mathematics, 103" (1983), Academic Press, Inc
- K. Nomizu, "Some results in E. Cartan's theory of isoparametric families of hypersur-faces", Bull. Amer. Math. Soc, 79, pp.1184~1188 (1973)[DOI]
- K. Nomizu, "Cartan's work on isoparametric families of hypersurfaces", Amer. Math. Soc., pp.191~200 (1975)