Mapped null hypersurfaces and Legendrian maps

For an (m+1)(m+1)-dimensional space–time (Xm+1,g), define a mapped null hypersurface to be a smooth map ν:Nm→Xm+1 (that is not necessarily an immersion) such that there exists a smooth field of null lines along νν that are both tangent and gg-orthogonal to νν.We study relations between mapped null hypersurfaces and Legendrian maps to the spherical cotangent bundle ST∗MST∗M of an immersed spacelike hypersurface μ:Mm→Xm+1. We show that a Legendrian map λ˜:Lm−1→(ST∗M)2m−1 defines a mapped null hypersurface in X. On the other hand, the intersection of a mapped null hypersurface ν:Nm→Xm+1 with an immersed spacelike hypersurface μ′:M′m→Xm+1 defines a Legendrian map to the spherical cotangent bundle ST∗M′ST∗M′. This map is a Legendrian immersion if νν came from a Legendrian immersion to ST∗MST∗M for some immersed spacelike hypersurface μ:Mm→Xm+1.