Mapped null hypersurfaces and Legendrian maps

Abstract

For an ( m + 1 ) -dimensional space–time ( X m + 1 , g ) , define a mapped null hypersurface to be a smooth map ν : N m → X m + 1 (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 S T ∗ M of an immersed spacelike hypersurface μ : M m → X m + 1 . We show that a Legendrian map λ ˜ : L m − 1 → ( S T ∗ M ) 2 m − 1 defines a mapped null hypersurface in X . On the other hand, the intersection of a mapped null hypersurface ν : N m → X m + 1 with an immersed spacelike hypersurface μ ′ : M ′ m → X m + 1 defines a Legendrian map to the spherical cotangent bundle S T ∗ M ′ . This map is a Legendrian immersion if ν came from a Legendrian immersion to S T ∗ M for some immersed spacelike hypersurface μ : M m → X m + 1 .