= Hypersurface orthogonality
{title2=$n\wedge dn=0$}
A nowhere-zero <differential one-form> $n$ is hypersurface orthogonal if locally $n=F\,dt$ with $F\ne0$. Its kernel consists of tangent vectors to the level hypersurfaces of $t$. <Frobenius theorem> makes this equivalent to $n\wedge dn=0$, where $d$ is the <exterior derivative>. With a torsion-free <Levi-Civita connection>, this is $n_{[\alpha}\nabla_\beta n_{\gamma]}=0$. For a timelike <unit normal>, projecting the derivative on both indices gives zero antisymmetric part, which explains why <hypersurface orthogonality implies symmetric extrinsic curvature>.
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