Conformal reference frames for Lorentzian manifolds

The twistor approach to Lorentzian manifolds (spacetimes) is based on the 5-dimensional space $\mathfrak{N}$ of all null geodesics. For the Minkowski space case it is described as “null projective twistors” (a real hypersufrace in the complex projective 3-space), but there is no canonical complex structure in general, curved case. One can rely on skies $\mathfrak{S}_x$ instead for each $x\in X$. The very definition of $\mathfrak{N}$ is problematical, unless the spacetime $X$ meets some convexity conditions. We can resort to the foliation of null geodesics in the bundle of skies, whereas $\mathfrak{N}$ (which could be defined as the space of leaves of the former) is endowed with a contact structure. This structure can be presented in terms of complex linear bundles and the Lorentz vector representation $({^1\!/\!_2},{^1\!/\!_2})$.
We shall consider a conformal reference frame, that is, a projection of the bundle of skies (as well as of $\mathfrak{N}$) onto a 3-dimensional real manifold, compatible with aforementioned contact structure. It permits to use surfaces (sky images) in the 3-manifold to describe the spacetime, as well as the differential geometry on it. A kind of “partial” almost complex structure on the bundle of skies appears, that compensates us for the loss of the global twistors’ complex structure.