In geometric quantization, one of the important ingredients is an integrable distribution $D$ (let's say real) on some manifold $M$ (symplectic, but this is not important). The resulting object is $M/D$, the space of all maximal integral submanifolds of $M$ with respect to $D$.


Set-theoretically, given that through each point of $M$ there passes a single maximal such submanifold, there is a natural surjection $\pi : M \to M/D$. Geometric quantization is interested only in those $(M, D)$ such that there exist a smooth structure on $M/D$ making $\pi$ a submersion. My question is: is it possible to have multiple non-diffeomorphic such smooth structures on $M/D$, or is there a single one (and then $M/D$ could the be considered as "the" quotient space of $M$ by $D$)?



Non-diffeomorphic smooth structures on the quotient of a manifold by an integrable distribution
Assuming there is a smooth structure on $M/D$ such that $\pi$ is a submersion. Then a function $f\colon M/D\to\mathbb R$ is smooth if and only if $f\circ\pi\colon M\to\mathbb R$ is smooth. This means, a homeomorphism $\psi\colon U\to V\subset\mathbb R^n$ for $U\subset M/D$ open (in the quotient topology) is a smooth chart for $M/D$ if and only if for all functions $g\colon V\to\mathbb R$, the composition $g\circ\psi\colon U\to\mathbb R$ is smooth if and only if $g$ is smooth. So the smooth structure is indeed unique if it exists. 


