TITLE: What is a submersion?
QUESTION [6 upvotes]: Don't misunderstand me, I read the Wikipedia article and I understood the definition - i.e. a function $f:\mathbb R^m\supseteq U\to\mathbb R^n$ is a submersion if $rk(f)\equiv n$, meaning that the differential is surjective everywhere.
I also know the connection to submanifolds, stating that a subset $M\subseteq\mathbb R^n$ is an $m$-dimensional $C^k$-submanifold if for every $p\in M$ there exists an environment $V\subseteq\mathbb R^n$ and a $C^k$-submersion $f:V\to\mathbb R^{n-m}$ such that $M\cap V=f^{-1}(0)$.
But what is a submersion? Is there any visual understanding of it? And how can one intuitively understand the connection to submanifolds?
I'm aware that this is not a mathematical precise question and hence am not expecting formally correct answers, I'm just trying to gain a deeper understanding to submersions.

REPLY [5 votes]: I think the answer you're looking for is very simple. The prototype of a submersion is the (linear) projection map $\pi\colon \Bbb R^m\to\Bbb R^n$ with $m\ge n$: Writing $\Bbb R^m = \Bbb R^n\times\Bbb R^{m-n}$, the map is given by $\pi(x,y)=x$. As an application of the inverse function theorem, you can prove that, any submersion $f\colon X\to Y$ ($X$, $Y$ manifolds—your $U$ and $\Bbb R^n$ if you like), in appropriate local coordinates on $X$ and $Y$ the mapping $f$ will be given exactly by $\pi$.