I'm related to my previous question here.
The problem is: I am given a $C^r$ manifold $M$ and a connected subset $A$ of $M$, and a $C^r$retraction $f:M\rightarrow A$ such that $f\vert_A=id:A\rightarrow A$. Then $A$ is a $C^r$ submanifold of $M$.
I have already shown that $f$ has constant rank in a neighbourhood $U$ in $M$ containing $A$. I must now use the Constant Rank Theorem, but I don't know how to. By it, there exists two diffeomorphisms, $$\alpha:U_1\underset{open}\subset U\rightarrow V\underset{open}\subset \mathbb{R}^k\times\mathbb{R}^{m-k}$$ and $$\beta:U_2\underset{open}\subset U\rightarrow W \underset{open}\subset \mathbb{R}^k\times\mathbb{R}^{m-k}$$ such that $\beta\circ f \circ \alpha^{-1}(x,y)=(x,0)$.
Now, what should be the next step to prove that $A$ is a submanifold? I tried working with the projection $$ \pi_2:\mathbb{R}^k\times\mathbb{R}^{m-k}, \pi_2(x,y)=y $$ but couldn't use the regular value theorem. Can someone help me?
I think your argument is complete:)
First of all, one of the definitions of submanifold is "locally image of immersion".
Secondly, regular value theorem is actually a special case of the constant rank theorem ($\beta$ is not needed in r.v.t. but actually it is a very simple function [no IFT magic, an honest formula]) and you don't need to use it (r.v.t. $\simeq$ c.r.t.) twice.
Just have a look: locally - in chart $\beta$ - the subset $A$ is a linear subspace of dimension $k$ (note that your formula [c.r.t] guarantees that $A\cap U_2$ is exactly this subspace) and since a linear subspace is obviously image of immersion (even a linear monomorphism), it is a submanifold.