The local embedding problem for optical structures (Q1802925)

The optical structure on a 4-dimensional manifold \(M\) has been introduced by \textit{I. Robinson} and \textit{A. Trautman} [Lett. Math. Phys. 10, 179-182 (1985)]. The author considers a Lorentzian metric and an optical structure on \(M\) and investigates the so-called shear-free geometry of rays. First he obtains a special framing adapted to a geometry of rays satisfying some equations which are similar to Cartan equations for Cauchy-Riemann manifolds. Then he proves that any \(M\) with a shear-free geometry of rays can locally be embedded in \(\mathbb{R} P^ 5\) in such a way that the induced structure and the intrinsic structure coincide.