Suppose we have a manifold $M$, and a connected submanifold $N$. We can make the quotient $\frac{M}{N}$, which send $N$ to a single point. Now, there are known restrictions on $N$ such that $\frac{M}{N}$ is also a manifold?
I can see that in the Euclidean space, as a trivial example, $\frac{\mathbb{R}^n}{\mathbb{R}^m}$ is $\mathbb{R}^{n-m}$ which is a manifold too ($m<n$). I guess this is closely related to the fact that $\mathbb{R}^m$ foliates $\mathbb{R}^n$ for any $m<n$. Is it necessary that $N$ foliates $M$? Is it sufficient?
I'm trying to answer the second question, which I think has a possitive answer, for which I'm using the definition of the foliation. Since any point $p$ in $M$ has a coordinate neighborhood and chart $(\phi, U)$ such that the connected components of $N \cap U$ are sets of the form {$q\in U | x^{m+1}(q)=a^{m+1}, \dots, x^{n}(q)=a^{n}$} for $a^{m+1}, \dots, a^{n}$ fixed real numbers, $m=$dim $N$ and $\phi=(x^1,\dots, x^n)$. So, locally our quotient looks like $R^{n-m}$, but I'm stuck with the Hausdorff property. In the first question, I've tried a little, but I don't even know whether it's possitive or not. Any help?