= Solution
Locally write a smooth <hypersurface> as $\Phi=0$ with $n_a=\nabla_a\Phi\ne0$. It is a <null hypersurface> when $n_an^a=0$ on it. Then $n^a$ is both normal and tangent, and the <induced metric> is degenerate along $n^a$.
Because <covariant derivatives> commute on a scalar,
$$
n^b\nabla_bn_a=n^b\nabla_an_b=\frac12\nabla_a(n_bn^b).
$$
The scalar $n^2$ vanishes on the <hypersurface>, so its derivative annihilates every tangent direction there. Its derivative is consequently proportional to $n_a$: locally $n^2=\Phi q$ for a smooth function $q$, giving
$$
\boxed{n^b\nabla_bn^a=\kappa n^a,\qquad \kappa=q/2\text{ on }\Phi=0.}
$$
Thus the normal generates <null pregeodesics>. Rescale $k^a=\alpha n^a$, choosing $n^b\nabla_b\log\alpha=-\kappa$, to get $k^b\nabla_bk^a=0$. These are the affinely parametrized generators of the <null hypersurface>. The proportionality need not vanish: setting $n^2=0$ only on the <hypersurface> does not set its full transverse derivative to zero.
Back to article page
