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 [51] 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].