Smooth foliation - Knowlpedia

A smooth foliation of dimension $p$ on an $n$ -manifold is an atlas of charts into open subsets of $\mathbb R^p\times\mathbb R^{n-p}$ whose transition maps preserve the local slices with fixed second coordinate. Equivalently, locally a transition has the form $(x,y)\mapsto(f(x,y),g(y))$ . The connected immersed submanifolds obtained by joining these slices are the leaves.

Leaf space

The quotient that identifies points on the same leaf is the leaf space. It can fail to be Hausdorff, even when the original manifold is compact and smooth. The irrational linear foliation of a torus is a basic example.