Frobenius theorem

Remarks

Condition 3 implies condition 2, since the slices $x^{k+1} = c_{k+1}, \dots, x^n = c_n$ are integral manifolds. Condition 2 implies condition 1, since vector fields tangent to a submanifold have a bracket tangent to it. The main step is that condition 1 implies condition 3. One picks local vector fields spanning $D$ that pairwise commute, which involutivity allows after a change of basis, and then uses their commuting flows as coordinates. Example: on $\R^3$ the distribution spanned by $X = \partial_x$ and $Y = \partial_y + x \, \partial_z$ is the kernel of $dz - x \, dy$. Here $[X, Y] = \partial_z$, which is not in $D$, so no surface is tangent to $D$; this is the standard contact structure. In the global form of the theorem, an involutive distribution splits $M$ into maximal connected integral manifolds, the leaves of a foliation.

Going deeper

For a student

The theorem turns a question about solving a system of first-order PDEs into an algebraic check on brackets. For $k = 1$ every line field is involutive, and the theorem reduces to the existence of integral curves of a vector field. A dual form uses differential forms: $D = \ker \theta^1 \cap \dots \cap \ker \theta^{n-k}$ is involutive if and only if each $d\theta^i$ lies in the ideal generated by the $\theta^j$. The theorem is used to integrate Lie subalgebras to Lie subgroups, to show that flat connections are locally trivial, and as the foundation of the theory of foliations.