Description
Foliation theory has its origins in the global analysis of solutions of ordinary differential equations: on an n-dimensional manifold M, an [autonomous] differential equation is defined by a vector field X ; if this vector field has no singularities, then its trajectories form a par tition of M into curves, i.e. a foliation of codimension n – 1. More generally, a foliation F of codimension q on M corresponds to a partition of M into immersed submanifolds [the leaves] of dimension ,——–,- – . – — p = n – q. The first global image that comes to mind is 1——–;- – – – – – that of a stack of “plaques”. 1———;- – – – – – Viewed laterally [transver 1——–1- – – — sally], the leaves of such a 1——–1 – – – – -. stacking are the points of a 1——–1— —-. quotient manifold W of di L….. -‘ _ mension q. —–~) W M Actually, this image corresponds to an elementary type of folia tion, that one says is “simple”. For an arbitrary foliation, it is only l- u L ally [on a “simpIe” open set U] that the foliation appears as a stack of plaques and admits a local quotient manifold. Globally, a leaf L may – – return and cut a simple open set U in several plaques, sometimes even an infinite number of plaques.
