My book is An Introduction to Manifolds by Loring W. Tu. Immersions and submersions are defined here.
- Let $A$ and $B$ be manifolds with the same dimension $d$, and let $G: A \to B$ be a smooth map. I think that for each $p \in A$, $G$ is a submersion at $p$ if and only if $G$ is an immersion at $p$ because $G_{*,p}$ is a homomorphism of vector spaces of the same finite dimension $d$.
Is this correct? If so, then I have 2 follow-up questions.
Can we restate Remark 8.12 of the inverse function theorem as follows?
$F$ is a local diffeomorphism at $p$ if and only if any of two equivalent conditions hold:
$F$ is a submersion at $p$,
$F$ is an immersion at $p$.
In this question What does it take for a smooth homeomorphism to be a diffeomorphism?, can we say submersion instead of immersion given that homeomorphism of smooth manifolds implies same dimension, as with diffeomorphism?
In some ways, I think one would expect immersion since what it takes for a smooth topological embedding to be a smooth embedding, as defined here, is being an immersion.
I was actually surprised to see immersion instead of submersion. Since submersions are open maps, I initially thought of submersion as the smooth analogue for "open map", in the sense that just as we have, for a bijective continuous map $g$ of topological spaces, that $g^{-1}$ is continuous if and only if $g$ is open, I thought that we would have, for the $f$ in the question, $f^{-1}$ is smooth if and only if $f$ is a submersion.
