Frobenius theorem
= Frobenius theorem
{c}
{wiki=Frobenius_theorem_(differential_topology)}
A constant-rank <smooth distribution> $D$ is an <integrable distribution> exactly when it is <involutive>. In differential-form language, let $\alpha^1,\ldots,\alpha^q$ be a local <basis> of its annihilator, so $D=\bigcap_i\ker\alpha^i$. The equivalent condition is
$$
d\alpha^i=\sum_j\beta^i{}_j\wedge\alpha^j
$$
for suitable local <one-forms> $\beta^i{}_j$. More precisely, the <ideal> generated by the annihilating forms inside the <exterior algebra> of <differential forms> is closed under the <exterior derivative>; this is an <exterior differential ideal>. It is not a claim that a two-form lies in the one-form annihilator itself.
For local <vector fields> $X,Y$ tangent to $D$, the definition of <exterior derivative> gives
$$
d\alpha(X,Y)=X(\alpha(Y))-Y(\alpha(X))-\alpha([X,Y])=-\alpha([X,Y]).
$$
Thus the displayed <closure> condition says exactly that the <Lie bracket of vector fields> remains tangent to $D$. The theorem turns this local algebraic condition into <integral manifolds> through every <point>.
For a nowhere-zero <one-form> $\alpha$, its kernel is tangent to local hypersurfaces precisely when $\alpha\wedge d\alpha=0$. In <general relativity>, applying this to a metric-dual one-form characterizes <hypersurface orthogonality> and vanishing twist.