Abstract
We extend the rigging technique to null submanifolds of a Lorentzian manifold, which allows us to construct two connections: the induced connection and the rigged connection. We study the relationship between them after changing the rigging and the existence and uniqueness of a preferred rigged connection, i.e., a rigged connection which coincides with the induced connection. As an application, we derive some integral formulas for compact null hypersurfaces with a preferred rigged connection and we obtain some consequences. For instance, we show that if the null mean curvature is zero, then the transverse null mean curvature vector field is orthogonal to any Killing vector field at some point.
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References
Atindogbé, C., Ezin, J.-P., Tossa, J.: Pseudoinversions of degenerate metrics. Int. J. Math. Math. Sci. 55, 3479–3501 (2003)
Atindogbé, C.: Scalar curvature on lightlike hypersurfaces. Appl. Sci. 11, 9–18 (2009)
Atindogbé, C., Fotsing, H.T.: Newton transformation on null hypersurfaces. Commun. Math. 23(1), 57–83 (2015)
Atindogbé, C., Gutiérrez, M., Hoummonkpe, R.: New properties on normalized null hypersurfaces. Mediterr. J. Math. 15, 166 (2018)
Duggal, K.L., Bejancu, A.: Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications. Kluwer Academic Publishers, Dordrecht (1996)
Duggal, K.L., Giménez, A.: Lightlike hypersurfaces of Lorentzian manifolds with distinguished screen. J. Geom. Phys. 55, 107–122 (2005)
Duggal, K.L., Sahin, B.: Differential Geometry of Lightlike Submanifolds. Birkhäuser Verlag, Basel (2010)
Gutiérrez, M., Olea, B.: Induced Riemannian structures on a null hypersurface. Math. Nachr. 289, 1219–1236 (2016)
Ngakeu, F., Fotsing, H.T.: \(\alpha -\)associated metrics in rigged null hypersurfaces. Turk J. Math. 43, 1161–1181 (2019)
Olea, B.: Null hypersurfaces on Lorentzian manifolds and rigging techniques. Spring. Proc. Math. Stat. 211, 237–251 (2016)
Yano, K.: Closed hypersurfaces with constant mean curvature in a Riemannian manifold. J. Math. Soc. Jpn. 17, 333–340 (1995)
Acknowledgements
The authors wish to acknowledge the funding support received for this publication from the Mentoring African Research in Mathematics (MARM) programme, which was sponsored by the London Mathematical Society and International Mathematical Union Committee for Developing Countries in partnership with the African Mathematics Millennium Sciences Initiative. The third author was 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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Benjamín Olea wishes to express his gratitude for the hospitality received during his visit to the Department of Mathematics of the University of Douala.
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Tetsing, H.F., Ngakeu, F. & Olea, B. Rigging Technique for 1-Lightlike Submanifolds and Preferred Rigged Connections. Mediterr. J. Math. 16, 139 (2019). https://doi.org/10.1007/s00009-019-1423-x
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DOI: https://doi.org/10.1007/s00009-019-1423-x