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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.

  1. 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 $p$.

  2. 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.

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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.

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  • $\begingroup$ Oh right so now that I know that homeomorphisms like diffeomorphisms imply the same dimension, we can simply replace immersion for submersion? Thanks! $\endgroup$ – user636532 Jul 20 '19 at 12:16
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    $\begingroup$ @SeleneAuckland Yup, given a smooth map between equal dimension manifolds, any time you see “immersion” feel free to say “submersion” or “local diffeomorphism” instead. $\endgroup$ – Santana Afton Jul 20 '19 at 12:22

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