Institute for Radiophysics and Electronics (IRE), NAS of Ukraine,

12 Ak. Proskura Str., 61085, Kharkiv, UKRAINE

E-mail: kost@ire.kharkov.ua

**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.