Null hypersurfaces in Lorentzian manifolds with the null energy condition

TL;DR

This paper proves that in certain complete Lorentzian manifolds satisfying the null energy condition, all properly immersed null hypersurfaces are necessarily totally geodesic, revealing a rigidity property of such hypersurfaces.

Contribution

It establishes a new rigidity result for null hypersurfaces in null energy condition Lorentzian manifolds, showing they must be totally geodesic if properly immersed.

Findings

Null hypersurfaces are totally geodesic in the given setting.

Properly immersed null hypersurfaces exhibit rigidity properties.

The result applies to geodesically complete Lorentzian manifolds with null energy condition.

Abstract

Let $M^{n+1}_1$ be a light-like geodesically complete Lorentzian $(n+1)$-manifold satisfying the null energy condition. We show that null hypersurfaces properly immersed in $M^{n+1}_1$ are totally geodesic.