 Solution

Affine geodesic motion gives $U^c{\nabla }_c{U}_a=0$. Commuting the two covariant derivatives and applying the product rule yieldsThis proves the required transport identity. The curvature term is symmetric after screen projection, so taking the antisymmetric screen part and inserting the optical decomposition gives the null-twist propagation equationIn particular, an initially twist-free congruence remains twist-free.