Intuitively speaking, a foliation of a manifold M is a decomposition of M into immersed submanifolds, the leaves of the foliation. These leaves are required to be of the same dimension, and to fit together nicely.

Such foliations of manifolds occur naturally in various geometric contexts, for example as solutions of differential equations and integrable systems, and in symplectic geometry. In fact, the concept of a foliation first appeared explicitly in the work of Ehresmann and Reeb, motivated by the question of existence of completely integrable vector fields on three-dimensional manifolds. The theory of foliations has now become a rich and exciting geometric subject by itself, as illustrated be the famous results of Reeb (1952), Haefliger (1956), Novikov (1964), Thurston (1974), Molino (1988), Connes (1994) and many others.

We start this book by describing various equivalent ways of defining foliations. A foliation on a manifold M can be given by a suitable foliation atlas on M, by an integrable subbundle of the tangent bundle of M, or by a locally trivial differential ideal. The equivalence of all these descriptions is a consequence of the Frobenius integrability theorem. We will give several elementary examples of foliations. The simplest example of a foliation on a manifold M is probably the one given by the level sets of a submersion M → N. In general, a foliation on M is a decomposition of M into leaves which is locally given by the fibres of a submersion.

In this chapter we also discuss some first properties of foliations, for instance the property of being orientable or transversely orientable.