Distributions and Foliations
Given a nonvanishing vector field on a smooth manifold M, the results of Chapter 9 show that the integral curves of the vector field fill up M and fit together nicely like parallel lines in Euclidean space. In this chapter we explore an important generalization of this idea to higher-dimensional submanifolds. Given a smooth subbundle of the tangent bundle of M, called a distribution on M, we can ask whether there are k-dimensional submanifolds (called integral manifolds of the distribution) whose tangent spaces at each point are the given subspaces of the tangent bundle. The answer in this case is more complicated than in the case of vector fields: there is a nontrivial necessary condition, called involutivity, that must be satisfied by the distribution. The main theorem of this chapter, the Frobenius theorem, tells us that involutivity is also sufficient for the existence of an integral manifold through each point. At the end of the chapter, we give applications of the theory to Lie groups and to partial differential equations.