Fibration in integral manifolds

Question

Consider a smooth manifold $M$ of dimension $n$ and an integrable tangent distribution $$ \mathcal{D} = span\{X_1,...,X_k\}$$ with $k\leq n$. Then we know that $M$ is foliated by the connected components of the integral manifolds of $\mathcal{D}$. Is there some condition on $\mathcal{D}$ which ensures that this foliation is indeed a fibration? If it may help, the generators of the distribution I have are actually Hamiltonian vector fields and $M$ is a symplectic manifold.

The only condition I know is that when these leaves are even the fibers of a proper and subjective submersion, then they are even fibers of a fibration (Ehresmann's Theorem). But, is there a way to directly say it is a fibration without involving a submersion?

Precisely, my problem is to see when quotienting $M$ with respect to this foliation I get that the leaf space is a manifold.

Answer 1

If the codimension of the foliation is one and the $M$ is closed, then $M$ fibers over $S^1$ if the foliation is defined by a closed non-vanishing one form. This is due to Tischler, see https://core.ac.uk/download/pdf/82577795.pdf

I am not sure if there are some related statements in higher codimension, though if there are some I think they come with some additional requirements.

For codimension one foliations one also has the Reeb stability theorem, which states that if the foliation is transversely orientable on a compact connected manifold and the foliation admits a compact leaf $L$ with finite homotopy group then the foliation is a given by a fiber bundle over $S^1$ with fiber $L$.

In general if the foliation is given by compact leaves with finite holonomy then one obtains an orbifold structure for the leaf space. An orbifold is a topological space which is locally modelled by some $\mathbb{R}^n$ modulo some action of a finite group. If all compact leaves have trivial holonomy then the leaf space is actually a manifold. A nice place to look into this would be for example the book: Introduction to foliation and Lie groupoids by Moerdijk and Mrcun.

You might also want to have a look at: https://mathoverflow.net/questions/186788/when-does-a-leaf-space-admit-a-non-hausdorff-manifold-structure

Answer 2

Here is another case where the foliation is in fact a submersion. You probably already know this one, but it's worth writing in case it helps someone.

Let $M$ be a manifold with a free smooth and proper $G-$action. Then the orbits of the action form a foliation $\mathcal{F}$ parametrized by the quotient space $M/G$. By the Quotient Manifold Theorem (see Lee's Smooth Manifolds Theorem 21.10) the map $M\to M/G$ realizes this foliation as a fibration with fibre $G$ such that $B=M/G$ is a smooth manifold with $\dim B=\dim M-\dim G$.

For instance given a Hamiltonian $G-$action on $(M,\omega)$ a symplectic manifold with moment map $\Phi:M\to \mathfrak{g}^*$, suppose $\mu$ is a regular value of $\Phi$ and $G_\mu$ acts freely and properly (properness is automatic for compact $G$) on $\Phi^{-1}(\mu)$.

Let $i_\mu:\Phi^{-1}(\mu)\to M$ denote the inclusion. It can be proven that $\ker(i_\mu^*\omega)|_p=T_p(G_\mu \cdot p)$ for $p\in \Phi^{-1}(\mu)$. In particular, writing $\sigma=i_\mu^*\omega$, since $\iota_{[X,Y]}=\mathcal{L}_Y\iota_X\sigma -\iota_Y\mathcal{L}_X\sigma$, if $X$ and $Y$ are vector fields valued in the distribution $\mathcal{D}_p:=\ker(i_\mu^*\omega)|_p$, so is $[X,Y]$.

Hence, $\mathcal{D}$ is an integrable distribution with leaves $G_\mu \cdot p$. It follows that $\Phi^{-1}(\mu)/G_\mu$ is a smooth manifold called the symplectic reduction of $(M,\omega)$ at $\mu$.