the Frobenius integrability theorem

Imagine at every point of a manifold you are handed a little k-dimensional plane of allowed directions — a field of tangent k-planes, called a distribution. The question is whether you can knit these planes together into surfaces that are tangent to them everywhere, like assembling a fabric whose threads everywhere point along the prescribed planes. Frobenius's theorem says exactly when this is possible, and the answer is a clean bracket condition.

A rank-k distribution D assigns to each p a k-dimensional subspace D_p of T_p M, varying smoothly. An integral manifold is an immersed k-submanifold whose tangent space at every point is exactly D_p. The distribution is called involutive if it is closed under the Lie bracket: whenever X and Y are vector fields lying in D (so X(p), Y(p) in D_p everywhere), the bracket [X, Y] also lies in D. Frobenius's theorem states that D is integrable — through every point passes an integral manifold, and locally the manifold is foliated by them into a stack of coordinate slices — if and only if D is involutive. In adapted coordinates the leaves become the slices x^{k+1} = const, ..., x^n = const.

Why the bracket condition? Because the bracket of two tangent fields measures the failure of their flows to commute, and if you could stay inside a surface tangent to D you would never be pushed out by such commutators — so involutivity is precisely the obstruction vanishing. This is the geometric heart of integrability and the reason it generalizes the existence of solutions to overdetermined first-order PDE systems. Honest caveat: involutivity is not automatic — a generic distribution is non-integrable. The standard contact distribution ker(dz - y dx) on R^3 is maximally non-integrable; no surface is tangent to it, precisely because the relevant bracket is everywhere transverse to the planes.