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.
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 $\mathcal{C}^{\infty}(M)$-submodule of $\mathfrak{X}(M)$ generated by $\{X_1,\cdots,X_r\}$. We will denote it by $\Xi_F = \Gamma(M,D)$.
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.
Under construction
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Author of the notes: Antonio J. Pan-Collantes
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