My question is about the use of the "local slice criterion" (as presented in Lee's "Introduction to Smooth Manifolds") to obtain an embedded submanifold from some subset $S$ of a manifold and the relationship between an embedded submanifold and a local embedding.
Specifically, I have a smooth manifold $M \subset R^n$ of dimension $m$ and a smooth map $\pi: M \rightarrow R^k$, where $k < m$. Say I know that, on $M$, $\pi$ has constant rank $r \leq k$. I then apply the constant rank theorem to infer, for every $p \in M$, the existence of smooth charts $(U, \phi)$ for $M$ centered at $p$, and $(V, \psi)$ for $R^k$ centered at $y = \pi(p)$ such that $\pi (U) \subset V$ and, in $U$, $\pi$ has the coordinate representation
$\tilde{\pi} (x^1,\ldots,x^{r},x^{r+1},\ldots,x^{m}) = (x^1,\ldots,x^{r},0,\ldots,0)$.
Let now $\epsilon > 0$ be so small that $\phi(U)$ contains a coordinate ball $U_\epsilon$ of radius $\epsilon$ centered at $\phi(p) = (p^1,\ldots,p^{r},p^{r+1},\ldots,p^m)$ and $\psi(V)$ contains a coordinate ball $V_\epsilon$ centered at $\psi(y) = (p^1,\ldots,p^{r},0,\ldots,0)$ . Setting $U_0 = \phi^{-1}(U_\epsilon)$ and $V_0 = \psi^{-1}(V_\epsilon)$ we have $p \in U_0 \subset U$ and $y \in V_0 \subset V$; furthermore, $\pi(U_0) \subset V_0$ is a local slice of $V_0$. Now, consider the subset $\pi(U_0) \subset R^k$: By the "local slice criterion" it seems to follow that $\pi(U_0)$ can then be made into an $r$-dimensional embedded submanifold of $R^k$.
My questions are: Is this a correct application of the criterion and, if so, then does it follow we can say that $\pi$ is a local embedding in $R^k$? What is bothering me is that it seems $\pi$ would then be an immersion and I don't think it is. I know this sounds like I haven't thought things through enough, but the only way I can reconcile all this is to say, roughly, that just because the image of a smooth map is an embedded submanifold does not make the map an embedding.
Any pointers would be much appreciated.
