A real distribution on a real smooth manifold is a vector subbundle of the tangent bundle . A complex distribution is a complex vector subbundle of the complexified tangent space of . A distribution of hyperplanes is a distribution of codimension in ; a distribution of complex hyperplanes is a distribution of complex codimension in .
One class of examples come from smooth foliations by submanifolds of constant dimension . Then the tangent vectors at all points to the submanifolds forming the foliation form a distribution of subspaces of dimension . The distributions of that form are said to be integrable.
say something about Frobenius theorem