My book is An Introduction to Manifolds by Loring W. Tu.
Let Let $N$ and $M$ be smooth manifolds of respective dimensions $n$ and $m$. Let $F:N \to M$ be a smooth map. Let $i: F(N) \to M$ be inclusion map of topological spaces. Let $\tilde F$ be $F$ with restricted range onto its image, i.e. $\tilde F: N \to F(N)$, $F = i \circ \tilde F$, a map of topological spaces. Theorem 11.15 tells us that if $F(N)$ is a (regular/embedded) $k$-submanifold, then $\tilde F$ and $i$ are smooth maps of manifolds.
Under the definition of smoothness between non-manifolds defined based on Definition 22.1 and Remark 22.5, is this converse of Theorem 11.15 true?
$F(N)$ is a (regular/embedded) submanifold if both $F$ and $\tilde F$ is smooth
- (assuming $F(N)$ is a manifold or assuming $F(N)$ is an immersed submanifold, depending on how much time and effort you're willing invest in answering this question)
What I've tried so far:
Assuming we have a definition of smoothness, for proving $F(N)$ is a regular submanifold (instead of embedded), I think we can just reverse the proof. Previously, an adapted chart was given. Now we obtain the adapted chart.
Assuming we have a definition of smoothness, for proving $F(N)$ is an embedded submanifold (instead of regular), I think we could try to prove $F$ or $i$ is an embedding.
For the definition of smoothness, I think it is: Let $p \in S \subseteq N$. The map $F: S \to M$ is smooth at $p$ if there is a neighborhood $U_p$ of $p$ in $N$ and a smooth map $g_p: U_p \to M$, where $g_p|_{U_p \cap S} = F|_{U_p \cap S}$
The problem is that the non-manifold, $S$, in the definition is in the domain. I don't think there's a definition for when the non-manifold is in the range.
Update: I found it on Wikipedia. Now, I can think about this, along with comments below, more and improve this question, or even answer, later.
Context: I believe we have $F$ immersion if and only if $\tilde F$ immersion, assuming $F(N)$ is a submanifold of $M$. If we assume $F(N)$ is only a manifold subset, then I think $\tilde F$ and $i$ are not necessarily smooth, so it doesn't make sense to talk about their differentials and thus whether or not they are immersions. This question then is to try to see if there's a chance of making sense of "$F$ immersion if and only if $\tilde F$ immersion" assuming only that $F(N)$ is a manifold subset. I guess there will be no chance but just making sure.