definition of foliation in manifold and why foliation is useful?

Question

I am thinking for the simple definition of "foliation" for a manifold. Why foliation is useful in manifold theory?

Answer 1

Roughly speaking, a codimension $n-q$ foliation $F$ on an $n$-manifold $M$ is partition of $M$ in $q$-manifolds, called leaves, such that locally $M$ is a product $R^{q}\times R^{n-q}$. Foliations are useful because they can give information about the topological structure of the manifold. For example a non-singular foliation on a 2-manifold $M$ implies that $M$ is the torus or the Klein bottle. A special case of a foliation is a non-singular flow, which serves as model for some physical systems.