Given a smooth manifold $M$ and an integrable distribution $\mathcal D$ on its tangent bundle $TM$. i.e $[\mathcal D,\mathcal D]\subset \mathcal D$. Then Froubenius theorem implies that $M$ is locally foliated by submanifolds whose tangent bundles are $\mathcal D$.
I don't understand what is the difference between locally foliated and globally foliated? Are they equivalent?
(We assume that $\mathcal{D}$ has constant rank) Recall that a distribution $\mathcal{D}\subset TM$ is integrable if there is a submanifold $N\subset M$ such that $T_pN=\mathcal{D}$ for all $p\in M$. Locally, $M$ can be written as $U=V\times F$, where $U\subset M$, $V\subset N$, and $F$ are all open. If we define the projection $\pi:M\rightarrow N$, then $(M,N,\pi,F)$ has the structure of a smooth fiber bundle. Note that it is not the case that $M=N\times F$ globally, just locally. A foliation on $TM$ is global if $M$ is diffeomorphic to $N\times F$, i.e. when this fibre bundle structure is flat. Otherwise the foliation is local.