Frobenius Integrability Theorem

The Frobenius integrability theorem states that a smooth constant-rank distribution on a manifold is tangent to a local foliation by integral manifolds precisely when it is an involutive distribution. Here tangent means that the tangent space of each leaf at a point equals the subspace selected there by the distribution. Equivalently, if two local vector fields and are tangent to the distribution, then their Lie bracket must also be tangent to it. The theorem therefore turns this differential condition into the existence of local solution manifolds.