4 - Foliations and Connections

Summary

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.