For manifolds of the same dimension, are submersions equivalent to immersions?

Question

My book is An Introduction to Manifolds by Loring W. Tu. Immersions and submersions are defined here.

1. 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.

2. 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 $p$.

3. 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.

Answer 1

You are correct on all three points.

The differential is a map between tangent spaces. If both tangent spaces have the same (finite) dimension, then an injective map is also a surjective map and is thus an isomorphism.

A local diffeomorphism between manifolds of the same dimension is indeed just an immersion or a submersion, as injectivity, surjectivity, and being an isomorphism on the level of tangent spaces are all equivalent.

If we have a smooth homeomorphism, your linked answer shows that it is a diffeomorphism if and only if it is an immersion. We know that a homeomorphism must be a map between manifolds of the same dimension, so here immersion is equivalent to submersion.