A subbundle of the tangent bundle of a manifold assigns to each point of the manifold a subspace of the tangent space . An integrable manifold of the subbundle is a submanifold of whose tangent space at each point in is . The subbundle is said to be integrable if for each point in , there is an integrable manifold of passing through .
By the Frobenius theorem, often proven in a first-year graduate course, a subbundle of the tangent bundle is integrable if and only if its space of sections is closed under the Lie bracket.
The Pontryagin ring of a vector bundle over is defined to be the subring of the cohomology ring generated by the Pontryagin classes of the bundle . In  Bott found an obstruction to the integrability of in terms of the Pontryagin ring of the quotient bundle . More precisely, if a subbundle of the tangent bundle is integrable, then the Pontryagin ring vanishes in dimensions greater than twice the rank of .
What is so striking about this theorem is not only the simplicity of the statement, but also the simplicity of its proof. It spawned tremendous developments in foliation theory in the Seventies, as recounted in [C] and [H1].