Foliation

A foliation $F$ is a collection of submanifolds (referred to as leaves) of a manifold $M$ satisfying certain conditions. They can be defined in many ways, one of which is by demanding the local existence of submersions from $U\subseteq M$ to ${\mathbb{R}}^{n-r}$.

Related: flag of foliations.

Regular ones

They can be characterized locally by the existence of an involutive distribution. That is, an involutive distribution gives rise, locally, to a foliation (the leaves would be the integral submanifold); and conversely: given a foliation, the tangent space determines a distribution.

The information needed to define the foliation corresponding to the distribution $D=\mathcal{S}(\{X_1,\cdots ,X_r\})$ is found in the ${\mathcal{C}}^{\mathrm{\infty }}(M)$-submodule of $\mathfrak{X}(M)$ generated by $\{X_1,\cdots ,X_r\}$. We will denote it by ${\mathrm{\Xi }}_F=\mathrm{\Gamma }(M,D)$.

The study of the space of leaves (which could be a manifold or not) is referred to as transverse geometry, with projectable vector fields playing a role here.

Singular ones

Under construction