Let $M$ be an $n$-manifold, and assume that it is foliated by a regular $p$-foliation. I know the following implication to be true:
If for every point $m\in M$ there exists a submanifold $m\in S\subset M$ which intersects every leaf at most once and which is such that for every $x\in S$ we have $T_xM = T_xS\oplus T_xF$ (where $F$ denotes the leaf through $x$), then the moduli space of leaves is a manifold in a natural way.
By the moduli space of leaves I mean $M/\sim$, where $x\sim y$ whenever $x$ and $y$ are on the same leaf. The proof goes by using the local slices $S$ as charts for the moduli space.
My question is: is the converse also true?
I believe it is not, but I have not been able to find a counterexample as of yet. Any help would be greatly appreciated.
I guess it depends on exactly what you mean by "in a natural way." One way to interpret that phrase is to say that $M/{\sim}$ has a smooth manifold structure such that the projection $M\to M/{\sim}$ is a smooth submersion. In this case, the converse is true: The rank theorem guarantees that in a neighborhood of each point $m\in M$, there exist smooth coordinates on $M$ and smooth coordinates on $M/{\sim}$ in which the projection has the coordinate representation $(x^1,\dots,x^k,x^{k+1},\dots,x^n)\mapsto (x^1,\dots,x^k)$. You can take $S$ to be the submanifold defined by $x^{k+1}=\dots=x^n=0$.