Mapped Null Hypersurfaces and Legendrian Maps

For an (m+1)-dimensional space-time (X, g), define a mapped null hypersurface to be a smooth map ν : N → X (that is not necessarily an immersion) such that there exists a smooth field of null lines along ν that are both tangent and g-orthogonal to ν. We study relations between mapped null hypersurfaces and Legendrian maps to the spherical cotangent bundle ST M of an immersed spacelike hypersurface μ : M → X. We show that a Legendrian map e λ : L → (ST M) defines a mapped null hypersurface in X. On the other hand, the intersection of a mapped null hypersurface ν : N → X with an immersed spacelike hypersurface μ : M → X defines a Legendrian map to the spherical cotangent bundle ST M. This map is a Legendrian immersion if ν came from a Legendrian immersion to ST M for some immersed spacelike hypersurface μ : M → X. We work in the C category, and the word “smooth” means C. The manifolds in this work are assumed to be smooth without boundary. They are not assumed to be oriented, or connected, or compact unless the opposite is explicitly stated. In this work (X, g) is an (m + 1)-dimensional Lorentzian manifold that is not assumed to be geodesically complete. A “vector field” on a manifold Y is a smooth section of the tangent bundle τY : TY → Y , and a “vector field along a map” φ : Y1 → Y2 of one manifold to another is a smooth map Φ : Y1 → TY2 such that φ = τY2 ◦ Φ. Covector fields and line fields on a manifold and along a map φ are defined in a similar way.