Update: Based on this, I think: Surjective immersions are (surjective) local diffeomorphisms because surjective immersions have $\dim {\text{domain}} = \dim{\text{range/image}}$. In particular, surjective immersions are equivalent to surjective local diffeomorphisms. Similarly, immersions whose images are (regular/embedded) submanifolds of range are local diffeomorphisms onto their images because $\dim {\text{domain}} = \dim{\text{image}}$, i.e. $n=k$, as below. Please verify that this in fact answers these three questions:
Let $N$ and $M$ be smooth manifolds with respective dimensions $n$ and $m$. Let $p \in N$. Let $F: N \to M$ be an immersion at $p$. It can be shown $F$ is a local embedding at $p$ (and conversely).
Main questions: If $F$ is an immersion for each $p \in N$ and $F(N)$ is, not only an immersed submanifold of $M$, but actually a (regular/an embedded) $k$-submanifold of $M$, then what can we say (about $F$, $N$, $F(N)$, the restriction $\tilde F: N \to F(N)$ etc)? Or what about if $F$ is additionally surjective (then its image is of course a submanifold of itself...I guess)?
So far I have: (I think these don't use that $F$ is an immersion.)
A. The inclusion $\iota: F(N) \to M$ is a smooth topological embedding and immersion.
B. The map $\tilde F: N \to F(N)$, $F$ with restricted range that satisfies $F = \iota \circ \tilde F$ is smooth.
Some guide questions for the main questions:
C. Does $N$ have the same dimension as $F(N)$, i.e. is $n=k$ ?
D. I haven't thought of any other guide questions.
I'm asking because
I find that the assumption that $F(N)$ is a submanifold makes the proof that $F$ is a local embedding a lot simpler, assuming the proof is correct.
- Whether or not the proof is correct, I'm thinking that there's something more to this. Perhaps assuming $F(N)$ submanifold makes $F$ into something where it would be obvious that $F$ is indeed a local embedding, such as because $F$ is actually an embedding (I think $F$ isn't necessarily injective) or a local diffeomorphism (I think immersions are not necessarily local diffeomorphisms).
We might be able to characterize such immersions as equivalent to local diffeomorphisms onto their images.
But in this case I think this would mean that surjective immersions are (surjective) local diffeomorphisms. While this would be consistent with both that bijective immersions are diffeomorphisms and that bijective local diffeomorphisms are diffeomorphisms, I wasn't able to find anything about surjective immersions being local diffeomorphisms, but I think this is true if $n=m$ and even if not surjective (and immersions would be local diffeomorphisms onto their images if $n=k$, I guess).
In relation to (1), I think (both local diffeomorphisms and) local diffeomorphisms onto images are local embeddings.
Since immersions are equivalent to local embeddings, I guess a local embedding with a submanifold image is a local diffeomorphism onto image.
For such an $F$, we can say $F$ is an immersion if and only if $\tilde F$ is an immersion. I suppose we might say this even if $F(N)$ weren't a submanifold or maybe even if not a manifold as long as we have some kind of smoothness definition.