Abstract
Most important examples of null hypersurfaces in a Lorentzian manifold admit an integrable screen distribution, which determines a spacelike foliation of the null hypersurface. In this paper, we obtain conditions for a codimension two spacelike submanifold contained in a null hypersurface to be a leaf of the (integrable) screen distribution. For this, we use the rigging technique to endow the null hypersurface with a Riemannian metric, which allows us to apply the classical Eschenburg maximum principle. We apply the obtained results to classical examples as generalized Robertson–Walker spaces and Kruskal space.
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Alías, L.J., Cánovas, V.L., Rigoli, M.: Trapped submanifolds contained into a null hypersurface of de Sitter spacetime. Commun. Contemp. Math. 20, 30 (2018)
Atindogbe, C., Gutiérrez, M., Hounnonkpe, R.: New properties on normalized null hypersurfaces. Mediterr. J. Math. 15, 19 (2018)
Asperti, A., Dajczer, M.: Conformally flat Riemannian manifolds as hypersurfaces of the light cone. Can. Math. Bull. 32, 281–285 (1989)
Duggal, K.L., Giménez, A.: Lightlike hypersurfaces of Lorentzian manifolds with distinguished screen. J. Geom. Phys. 55, 107–222 (2005)
Duggal, K.L., Sahin, B.: Differential Geometry of Lightlike Submanifolds. Birkhäuser Verlag, Boston (2010)
Eschenburg, J.H.: Maximum principle for hypersurfaces. Manuscr. Math. 64, 55–75 (1989)
Fotsing Tetsing, H., Ngakeu, F., Olea, B.: Rigging technique for 1-lightlike submanifolds and preferred rigged connections. Mediterr. J. Math. 16, 20 (2019)
Gutiérrez, M., Olea, B.: Conditions on a null hypersurface of a Lorentzian manifold to be a null cone. J. Geom. Phys. 145, 9 (2019)
Gutiérrez, M., Olea, B.: Induced Riemannian structures on null hypersurfaces. Math. Nachr. 289, 1219–1236 (2016)
Gutiérrez, M., Olea, B.: Totally umbilic null hypersurfaces in generalized Robertson–Walker spaces. Differ. Geom. Appl. 42, 15–30 (2015)
Liu, H., Jung, S.D.: Hypersurfaces in lightlike cone. J. Geom. Phys. 58, 913–922 (2008)
Liu, J.L., Yu, C.: Rigidity of isometric immersions into the light cone. J. Geom. Phys. 132, 363–369 (2018)
Penrose, R.: Gravitational collapse and space-time singularities. Phys. Rev. Lett. 14, 57–59 (1965)
Palomo, F.J., Romero, A.: On spacelike surfaces in four-dimensional Lorentz–Minkowski spacetime through a light cone. Proc. R. Soc. Edinb. Sect. A 143, 881–892 (2013)
Palomo, F.J., Rodríguez, F.J., Romero, A.: New characterizations of compact totally umbilical spacelike surfaces in 4-dimensional Lorentz–Minkowski spacetime through a lightcone. Mediterr. J. Math. 11, 1229–1240 (2014)
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Communicated by Young Jin Suh.
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This paper has been partially supported by the Ministerio de Economía y Competitividad Grant FEDER-MTM2016-78647-P and Junta de Andalucía research group FQM-324.
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Gutiérrez, M., Olea, B. Codimension Two Spacelike Submanifolds Through a Null Hypersurface in a Lorentzian Manifold. Bull. Malays. Math. Sci. Soc. 44, 2253–2270 (2021). https://doi.org/10.1007/s40840-020-01056-w
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DOI: https://doi.org/10.1007/s40840-020-01056-w