Abstract
We introduce the concept of rigging for a null hypersurface in a Lorentzian manifold, which allows us to induce all the necessary geometric objects in a null hypersurface and also to define a Riemannian metric on it, called rigged metric. This metric can be used as an auxiliary tool to study the null hypersurface. Its Levi-Civita connection is called rigged connection and, in general, it will not coincide with the induced connection on the null hypersurface. We show a necessary and sufficient condition for this to happen and we give some examples. Since both rigged connection as induced connection depend on the rigging, we investigate if they can coincide for a suitable choice of the rigging.
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References
M.A. Akivis, V.V. Goldberg, On some methods of construction of invariant normalizations of lightlike hypersurfaces. Differ. Geom. Appl. 12, 121–143 (2000)
C. Atindogbe, Normalization and prescribed extrinsic scalar curvature on lightlike hypersurfaces. J. Geom. Phys. 60, 1762–1770 (2010)
C. Atindogbe, J.P. Ezin, T. Tossa, Pseudo-inversion of degenerate metrics. Int. J. Math. Math. Sci. 55, 3479–3501 (2003)
C. Atindogbe, K.L. Duggal, Conformal screen on lightlike hypersurfaces. Int. J. Pure Appl. Math. 11, 421–442 (2004)
K.L. Duggal, A. Giménez, Lightlike hypersurfaces of Lorentzian manifolds with distinguished screen. J. Geom. Phys. 55, 107–222 (2005)
M. Gutiérrez, B. Olea, Global decomposition of a Lorentzian manifold as a generalized Robertson-Walker space. Differ. Geom. Appl. 27, 146–156 (2009)
M. Gutiérrez, B. Olea, Induced Riemannian structures on a null hypersurface. Math. Nachr. 289, 1219–1236 (2016)
M. Gutiérrez, B. Olea, Totally umbilic null hypersurfaces in generalized Robertson-Walker spaces. Differ. Geom. Appl. 42, 15–30 (2015)
M. Navarro, O. Palmas, D.A. Solis, Null hypersurfaces in generalized Robertson-Walker spacetimes. J. Geom. Phys. 106, 256–267 (2016)
B. Olea, Canonical variation of a Lorentzian metric. J. Math. Anal. Appl. 419, 156–171 (2014)
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Olea, B. (2017). Null Hypersurfaces on Lorentzian Manifolds and Rigging Techniques. In: Cañadas-Pinedo, M., Flores, J., Palomo, F. (eds) Lorentzian Geometry and Related Topics. GELOMA 2016. Springer Proceedings in Mathematics & Statistics, vol 211. Springer, Cham. https://doi.org/10.1007/978-3-319-66290-9_13
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DOI: https://doi.org/10.1007/978-3-319-66290-9_13
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