Minimal null two-surfaces in 4D Lorentzian space-times

Abstract:
Observing a failure of standard methods of Riemannian geometry for description of null two-surfaces (two-surfaces with degenerate first fundamental form) in 4D Loretzian space-times, we propose a spinor limiting procedure, which turns out to be useful for formulating the notion of a minimal null two-surface. A study of geometry of minimal two-surfaces is presented. It shows that the geometry of a minimal null two-surface depends on whether the corresponding line of striction
is a null or space-like curve. In the former case the minimal null two-surface is (locally) developable, ruled by null geodesics, and the null geodesics of the congruence are strongly incident; in the latter case the null generators of the two-surface present an example of, what was called by Penrose, weakly incident light rays. These results exhibit unusual features connected with the indefinite property of the Lorentz norm.