 Solution

On the null hypersurface, the generator covector is proportional to a normal, locally $U_a=f{\nabla }_{a}u$ for a level-set function $u$. The Frobenius theorem therefore givesContracting once with $N^a$ and projecting the remaining indices with $P^a{}_b$ removes every term containing $U$ or $N$ and leaves $P_b{}^d{P}_c{}^e{\nabla }_{[d}{U}_{e]}=0$. Henceon the hypersurface.