In this chapter we introduce foliations and discuss some fundamental examples. We characterise the integrability of subbundles of tangent bundles in terms of both flatness and torsion-freeness of suitable affine connections. In the final section we discuss the simultaneous integrability of complementary distributions making up an almost product structure.

We introduce Bott connections in general, and we apply them to Lagrangian foliations in particular. This leads to a proof of Weinstein’s characterisation of affinely flat manifolds as leaves of Lagrangian foliations. We also prove a Darboux theorem for pairs consisting of a symplectic structure together with a Lagrangian foliation.

This book is about the geometry and topology of symplectic manifolds equipped with pairs of complementary Lagrangian foliations. The resulting structure is very rich, intertwining symplectic geometry, the theory of foliations, dynamical systems, and pseudo-Riemannian geometry in interesting ways.

Before describing the contents of the book in detail, we want to discuss a few motivating vignettes. The first two of these are to be kept in mind as motivational background, whereas the third and fourth ones will be taken up again and again later in the book.