  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.
