# Restriction of a submersion to a certain kind of submanifold

I'm trying to prove the following result: let $$f\colon M\to N$$ be a surjective submersion between two manifolds, and $$E\subseteq N$$ an embedded submanifold. Then $$S=f^{-1}(E)$$ is also an embedded submanifold of $$M$$ and $$f\colon S\to E$$ is a submersion.

So far I've been able to prove that $$S$$ is an embedded submanifold via local coordinates, but I'd be grateful if you could check my reasoning for proving that $$f\colon S\to E$$ is a submersion.

Let $$p\in S$$ and $$q=f(p)$$. Since $$f$$ is a submersion, there exists a local smooth section $$\sigma\colon V\subseteq N\to M$$ of $$f$$ such that $$p=\sigma(q)$$. Therefore, the restriction $$\sigma\colon E\cap V\to M$$ is smooth, and $$\sigma(E\cap V)\subseteq S$$, since for every $$z\in E\cap V$$ we have $$f(\sigma(z))=z\in E$$, so $$\sigma(z)\in S$$. Because of this, and the fact that $$S$$ is embedded, $$\sigma\colon E\cap V\to S$$ is a local section of $$f\colon S\to E$$ such that $$\sigma(q)=p$$, which implies that $$f$$ is a submersion at $$p$$.

Is my reasoning correct? Thank you in advance!

• It falls apart at $q=f(p)$. We have $p \in E \subset N$ which is the range of the function $f$, not the domain, so the evaluation $f(p)$ makes no sense. – Lee Mosher Aug 6 at 19:44
• Oh, my mistake. I meant $p\in S$. I'll change it now, thank you for noticing it. – Darth Lubinus Aug 6 at 20:22