Solution
= 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 gives
$$
U_{[a}\nabla_bU_{c]}=0.
$$
Contracting once with $N^a$ and projecting the remaining indices with $P^a{}_b$ removes every term containing $U$ or $N$ and leaves $P_b{}^dP_c{}^e\nabla_{[d}U_{e]}=0$. Hence
$$
\boxed{\widehat\omega_{ab}=0}
$$
on the hypersurface.