A (regular) foliation of codimension on a manifold is a partition of into immersed connected submanifolds of codimension such that
satisfying the local triviality property.
That is, for every there is an open neighborhood of such that
coincides with the partition by the fibers of a submersion
(Note that this means that .)
The submanifolds are called leaves of the foliation.
Informal explanation
Informally, a foliation is a way to break up a manifold into a bunch of submanifolds that are the same āsizeā and all behave similar enough so that each one gets sent to a certain point by a submersion.
Note that the definition given in Leeās book (@lee2013) of a foliation is a collection of disjoint, connected, nonempty, immersed submanifolds of whose union is and such that in a neighborhood of each point , there exists a flat chart for .
It is pretty easy to see these definitions are equivalent, but the one stated above is more detailed.