The Rigging Technique for Null Hypersurfaces

Round 1

Reviewer 1 Report

This survey is devoted to the rigging technique introduced by the authors in [38] (and then studied in several articles) for null hypersurfaces $i : L \to M$ in a Lorentzian manifold $(M, g)$. They assume existence of a vector field $\zeta$ given on a neighborhood of $L$ and transversal to $L$; this $\zeta$ is called a "rigging vector field". They consider the degenerate induced metric $i^*g$ on $L$ and define a Riemannian (rigging) metric $\tilde g=\omega\otimes\omega+i^g$ on $L$, where $\omega=i^\alpha$ and $\alpha$ (in a non-degenerate case) is the one-form dual to $\zeta$.  The authors show how rigging  can be used to study properties of null hypersurfaces, with emphasis in 4 null cones, totally geodesic, totally umbilic and compact null hypersurfaces. The article also considers applications to cosmological models. In Section 4 they discuss further developments of the rigging technique, say, for contact and Sasakian structures and the singularity theory. 

I see a misprint in line 1115: "Three kind ..." -> "Three kinds ...".

I think that the work can be published in the journal.

Reviewer 2 Report

The rigging technique for null hypersurfaces  in   Lorentzian manifolds  have  been introduced  in  a previous authors’  paper   “Induced Riemannian structures on null hypersurfaces”,  Mathematische Nachrichten, 289(2016), 1219–1236. The present review article gives an expose of properties of this technique. The first chapter is devoted to a motivation, a description of  two classical techniques to study a null hypersurface, and authors' contributions to a new  geometric point of view  for the same object.  The second section explains  the rigging technique in detail .  This includes  a rigged vector field and a rigged metric, a rigged connection,  and curvature  relations. A main role in this article plays the third section considering  applications.  The use  of the rigging technique to totally umbilic and totally geodesic null hypersurfaces,  null cones, completeness of the rigged metric , codimension two spacelike  submanifolds through a null hypersurface,  compact null hypersurfaces and   Black Hole horizons shows the importance of such a kind of research. In the last Section 4,  the main results of the presented theory are discussed. In conclusion, the paper is very well written and suitable for the  journal "Axioms".