Update: Based on this, I think: Surjective immersions are local diffeomorphisms because surjective immersions have $\dim {\text{domain}} = \dim{\text{range/image}}$. 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:
Question A. For these proofs (one on stackexchange, one on wordpress) that immersions are local embeddings:
I understand that images of immersions are immersed submanifolds, not necessarily (regular/embedded) submanifolds and not necessarily manifolds.
Do I understand correctly that, nevertheless, immersions are local embeddings and that these proofs indeed prove such with the images of the immersions being assumed only immersed submanifolds?
Question B. Is this proof for immersion, whose image is a submanifold, is a local embedding correct? (I won't prove that local embeddings are immersions.)
- In particular, I'm not sure about (5). I think (5) is true if $F(N)$ has the same dimension as $N$. I think I at least proved local diffeomorphisms onto images with images submanifolds are local embeddings
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 $p$. Let its image $F(N)$ be a (regular/embedded) $k$-submanifold of $M$.
Let us show $F$ is a local embedding at $p$, defined as that there exists a neighborhood $V_p$ of $p$ in $N$ such that $F|_{V_p}: V_p \to M$ is an embedding.
Because $F(N)$ is a submanifold of $M$, $F(N)$ is a manifold, so it would make sense to say, if we were to say that
3.1. The inclusion $\iota: F(N) \to M$ is a map of manifolds
3.2. $\tilde F: N \to F(N)$, $F$ with restricted range that satisfies $F = \iota \circ \tilde F$, is a map of manifolds.
- Assert (3.1) and (3.2). Additionally, the maps in (3) are smooth because $F(N)$ is a submanifold:
- $\tilde F$ is a local diffeomorphism, i.e. $F$ is a "local diffeomorphism onto its image" by this, where there is no ambiguity in the definition of "local diffeomorphism onto its image" because its image is a submanifold.
- Edit: Not quite sure about this step actually. I think true if $n=k$.
Let $G=\tilde F$ By (5), there exists a neighborhood $U_p$ of $p$ in $N$ such that $G(U_p)$ is open in $F(N)$, and $\tilde{G|_{U_p}}: U_p \to G(U_p)$ is a diffeomorphism, where $\tilde{G|_{U_p}}$ is $G|_{U_p}: U_p \to F(N)$ with restricted range.
Choose $V_p = U_p$. This works because, under this definition for embedding, equivalent to the more natural one:
7.1 $G|_{U_p}$ is an immersion if and only if $\tilde{G|_{U_p}}$ is an immersion, by this.
7.2 $G|_{U_p}$ is a topological embedding because $\tilde{G|_{U_p}}$ is a homeomorphism because $\tilde{G|_{U_p}}$ is a diffeomorphism.
7.3 $F|_{U_p} = \iota \circ G|_{U_p}$
7.4 $F|_{U_p}$ is an immersion if both $G|_{U_p}$ and $\iota$ are immersions.
7.5 $\iota$ is an immersion by this.
7.6 $\tilde{G|_{U_p}}$ is an immersion since $\tilde{G|_{U_p}}$ is a diffeomorphism.
7.7 $G|_{U_p}$ is an immersion by (7.1) and (7.6).
7.8 Therefore, $F|_{U_p}$ is an immersion by (7.4) and (7.7).
7.9 $F|_{U_p}$ is a topological embedding if and only if $G|_{U_p}$ is a topoloigcal embedding.
7.10 Therefore, $F|_{U_p}$ is a topological embedding by (7.9) and (7.2).
7.11 Therefore, $F|_{U_p}$ is a smooth embedding by (7.10) and (7.8).
