Weyl curvature as a cross-ratio of points on the celestial sphere

The Weyl curvature appears in Einstein’s field equations, and is one of the two main types of space-time curvature considered in general relativity. It is possible to view the Weyl curvature as a ridge system on a 2-sphere with index four singularities representing the principle null directions. I am interested in the cross ratio of these four singularities in terms of the Weyl curvature. This paper explores the mathematical foundation to look into this question.