On lightlike hypersurfaces and lightlike focal sets of null Cartan curves in Lorentz-Minkowski spacetime

In this paper, as a type of event horizons in astrophysics, a class of lightlike hypersurfaces that is generated by null curves will be investigated and discussed. Based on discussions of the properties of the local differential geometry of null curves and singularity theory, we provide classifications of the singularities of lightlike hypersurfaces and lightlike focal sets. In addition, we reveal the facts that the types of these singularities and the order of contact between a null Cartan curve and a pseudosphere are related closely to null Cartan curvatures. Finally, examples of lightlike hypersurfaces and lightlike focal set are used to demonstrate our theoretical results.