Null hypersurfaces as wave fronts in Lorentz-Minkowski space

TL;DR

This paper explores the geometric structure of null hypersurfaces in Lorentz-Minkowski space, revealing their canonical relation to Euclidean hypersurfaces and classifying those with compact singular sets.

Contribution

It establishes a canonical correspondence between null hypersurfaces as wave fronts in Lorentz-Minkowski space and Euclidean hypersurfaces, and classifies null wave fronts with compact singular sets.

Findings

Most null wave fronts are restrictions of $L$-complete null wave fronts.

Characterization of $L$-complete null wave fronts with compact singular sets.

Null hypersurfaces can be induced by Euclidean hypersurfaces.

Abstract

In this paper, we show that ``$L$-complete null hypersurfaces'' (i.e. ruled hypersurfaces foliated by entirety of light-like lines) as wave fronts in the $(n+1)$-dimensional Lorentz-Minkowski space are canonically induced by hypersurfaces in the $n$-dimensional Euclidean space. As an application, we show that most of null wave fronts can be realized as restrictions of certain $L$-complete null wave fronts. Moreover, we determine $L$-complete null wave fronts whose singular sets are compact.