= Frobenius theorem
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{wiki=Frobenius_theorem_(differential_topology)}
If $D=\bigcap_{i=1}^q\ker\alpha^i$ has constant rank, the Frobenius theorem says that $D$ is <integrable distribution>[integrable] exactly when its annihilator is closed under the <exterior derivative>: equivalently,
$$
d\alpha^i=\sum_j\beta^i{}_j\wedge\alpha^j.
$$
This is also equivalent to involutivity of $D$.
For a nowhere-zero 1-form $\alpha$, the hyperplane distribution $\ker\alpha$ is tangent to hypersurfaces exactly when $\alpha\wedge d\alpha=0$. In <general relativity> this characterizes hypersurface-orthogonal vector fields and implies vanishing twist.
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