Abstract
Lightlike Cartan geometries are introduced as Cartan geometries modelled on the future lightlike cone in Lorentz-Minkowski spacetime. Then, we provide an approach to the study of lightlike manifolds from this point of view. It is stated that every lightlike Cartan geometry on a manifold N provides a lightlike metric h with radical distribution globally spanned by a vector field Z. For lightlike hypersurfaces of a Lorentz manifold, we give the condition that characterizes when the pull-back of the Levi-Civita connection form of the ambient manifold is a lightlike Cartan connection on such hypersurface. In the particular case that a lightlike hypersurface is properly totally umbilical, this construction essentially returns the original lightlike metric. From the intrinsic point of view, starting from a given lightlike manifold (N, h), we show a method to construct a family of ambient Lorentzian manifolds that realize (N, h) as a hypersurface. This method is inspired on the Feffermann-Graham ambient metric construction in conformal geometry and provides a lightlike Cartan geometry on the original manifold when (N, h) is generic.
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Notes
I am grateful to Professor Cristina Draper for the indications for the proof of this Lemma.
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The author would like to give his sincere thanks to the referees for the careful reading of the manuscript and their corresponding suggestions.
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The author is partially supported by the Spanish MINECO and ERDF project MTM2016-78807-C2-2-P and by the Junta Andalucía and ERDF I+D+I project A-FQM-494-UGR18.
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Palomo, F.J. Lightlike manifolds and Cartan geometries. Anal.Math.Phys. 11, 112 (2021). https://doi.org/10.1007/s13324-021-00547-8
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DOI: https://doi.org/10.1007/s13324-021-00547-8