Just to be clear: the objects we're talking about here are smooth ($C^\infty$) manifolds (without a boundary) and submersion is defined as a map between manifolds, which has constant rank that is equal to the dimension of the codomain.
While trying to do an exercise I kept stumbling upon the idea that "submersion is locally a projection and therefore an open map". This idea is not the problem. The "problem" is the theorem that states the following:
Let $M$ and $N$ be smooth manifolds, $dim M=m$, $dim N=n$, and let $f:M\to N$ be a smooth mapping of a constant rank $r$. For every $p\in M$ there is a (smooth) chart $(U, \varphi)$ at $p$ and chart $(V, \psi)$ at $f(p)$, such that $f(U)\subset V$ and such that $f$ has coordinate representation $$\psi \circ f \circ \varphi ^{-1} (x_1,\ldots ,x_r,x_{r+1}, \ldots ,x_m)=(x_1,\ldots ,x_r,0,\ldots ,0)$$
Doesn't this mean that every constant rank mapping, not only submersions, is locally a projection? It seems to me that I've completely misunderstood the idea of "locally being a projection", because that should be something that's very characteristic of submersions. Also, if every constant rank mapping is locally a projection, that would mean there are no constant rank mappings from compact manifolds to euclidean space.
So, my question is: what do people mean when they say "submersion is locally a projection"?
Best Answer
This local form $(x_1,\ldots, x_r,x_{r+1},\ldots, x_m)\to (x_1,\ldots, x_r,0,\ldots,0)$ is written slightly misleadingly. If we examine a few special cases, we can see that these maps are not all "projections" in the sense which you intend. If $m\ge n=r$, we have that the map is of the form $$ (x_1,\ldots, x_m)\mapsto(x_1,\ldots, x_r)$$ and is indeed locally a bona fide projection. If $m\ge n>r$, then the map looks like $$ (x_1,\ldots, x_m)\mapsto (x_1,\ldots,x_r)\mapsto (x_1,\ldots, x_r,0,\ldots,0)$$ and hence a composition of a projection and then an inclusion. In the case where $r=m\le n$, this map becomes $$ (x_1,\ldots, x_m)\mapsto(x_1,\ldots, x_m,0,\ldots, 0)$$ which is an inclusion. If $r<m\le n$, we get $$ (x_1,\ldots, x_m)\mapsto (x_1,\ldots, x_m,0,\ldots,0)\mapsto (x_1,\ldots, x_r,0,\ldots, 0)$$ which is composition of an inclusion and a projection.
The moral of the story is that there are a few varied behaviors. For an example, take the inclusion of $S^2\hookrightarrow \Bbb{R}^3$. This is a constant rank $2$ map, so the theorem tells us that locally it looks like $(x_1,x_2)\mapsto (x_1,x_2,0)$. I.e. locally it is a standard inclusion. It is an example of an inclusion of a compact manifold into Euclidean space, and does not contradict the theorem.