Null hypersurfaces in 4-manifolds endowed with a product structure

TL;DR

This paper investigates null hypersurfaces in a special class of 4-manifolds with a neutral metric derived from an Einstein metric and an almost paracomplex structure, revealing conditions for their geometric properties.

Contribution

It provides new conditions for the existence and properties of null hypersurfaces in 4-manifolds with a conformally flat, scalar flat neutral metric.

Findings

Totally geodesic null hypersurfaces are scalar flat.

Existence of such hypersurfaces implies the ambient Einstein metric is Ricci-flat.

Necessary conditions are established for null hypersurfaces with constant mean curvature.

Abstract

In a 4-manifold, the composition of a Riemannian Einstein metric with an almost paracomplex structure that is isometric and parallel, defines a neutral metric that is conformally flat and scalar flat. In this paper, we study hypersurfaces that are null with respect to this neutral metric and in particular we study their geometric properties with respect to the Einstein metric. Firstly, we show that all totally geodesic null hypersurfaces are scalar flat and their existence implies that the Einstein metric in the ambient manifold must be Ricci-flat. Then, we find a necessary condition for the existence of null hypersurface with equal non-trivial principal curvatures and finally, we give a necessary condition on the ambient scalar curvature, for the existence of null (non-minimal) hypersurfaces that are of constant mean curvature.