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 $\nu :N^m\to X^{m+1}$ (that is not necessarily an immersion) such that there exists a smooth field of null lines along $\nu$ that are both tangent and $g$-orthogonal to $\nu .$ We study relations between mapped null hypersurfaces and Legendrian maps to the spherical cotangent bundle $S{T}^{\ast }M$ of an immersed spacelike hypersurface $\mu :M^m\to X^{m+1}.$ We show that a Legendrian map $\wt λ: L^{m-1}\to (ST^*M)^{2m-1}$ defines a mapped null hypersurface in $X.$ On the other hand, the intersection of a mapped null hypersurface $\nu :N^m\to X^{m+1}$ with an immersed spacelike hypersurface ${\mu }^{\prime }:M^{\prime m}\to X^{m+1}$ defines a Legendrian map to the spherical cotangent bundle $S{T}^{\ast }{M}^{\prime }.$ This map is a Legendrian immersion if $\nu$ came from a Legendrian immersion to $S{T}^{\ast }M$ for some immersed spacelike hypersurface $\mu :M^m\to X^{m+1}.$

13 pages, 1 figure