Are immersions equivalent to local diffeomorphisms onto their images if their images are submanifolds?

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:

My book is An Introduction to Manifolds by Loring W. Tu.

Let $$N$$ and $$M$$ be smooth manifolds with respective dimensions $$n$$ and $$m$$. Let $$p \in N$$. Let $$F: N \to M$$ be a smooth map. Assume further $$F(N)$$ is a (regular/embedded) $$k$$-submanifold of $$M$$.

Question A: Are these correct?

1. Then $$F(N)$$ is a manifold, so it would make sense to say, and I do say, that

• 1.1. The inclusion $$\iota: F(N) \to M$$ is a map of manifolds

• 1.2. $$\tilde F: N \to F(N)$$, $$F$$ with restricted range that satisfies $$F = \iota \circ \tilde F$$, is a map of manifolds.

• 1.3 Since our manifolds have dimensions, if we have that $$\dim N = \dim F(N)$$ for $$p \in N$$, then $$\dim N = \dim F(N)$$ at every $$q \in N$$.

2. Additionally, $$\iota$$ and $$\tilde F$$ are smooth because $$F(N)$$ is a submanifold by, respectively, this and this.

3. Given (1) and (2),

• 3.1. it also makes sense to say if we were to talk about whether or not $$\tilde F$$ is a local diffeomorphism at $$p$$ (equivalent to whether or not $$F$$ is a local diffeomorphism onto its image at $$p$$), an immersion at $$p$$, a submersion at $$p$$, etc.

• 3.2. If $$N$$ and $$F(N)$$ have the same dimension, then $$\tilde F$$ is a local diffeomorphism at $$p$$ if and only if $$\tilde F$$ is an immersion at $$p$$ if and only if $$\tilde F$$ is a submersion at $$p$$. (Even when you don't think of (1.3), I guess (3.2) may hold if you assume only "have the same dimension at $$p$$". I haven't thought about it.)

• 3.3. $$F$$ is an immersion at $$p$$ if and only if $$\tilde F$$ is an immersion at $$p$$ , by this, regardless of same dimension.

Question B: Is $$F$$ an immersion at p if $$\tilde F$$ is a local diffeomorphism at $$p$$, i.e. a local diffeomorphism onto its image at $$p$$? Here is my proof. Please verify.

• B1. There exists a neighborhood $$U_p$$ of $$p$$ in $$N$$ such that $$F(U_p)$$ is open in $$F(N)$$, and $$\tilde{F|_{U_p}}: U_p \to F(U_p)$$ is a diffeomorphism, where $$\tilde{F|_{U_p}}$$ is $$F|_{U_p}: U_p \to F(N)$$ with restricted range.

• B2. $$\dim N = \dim U_p = \dim F(U_p) = \dim F(N)$$ at $$p \in N$$, by (B1).

• B3. $$\dim N = \dim F(N)$$ at every $$q \in N$$ by (B2) and (1.3).

• B4. Therefore, $$F$$ is an immersion at $$p$$ by (B3), (3.2) and (3.3)

Question C: Let $$F$$ be an immersion at p. Is $$\tilde F$$ a local diffeomorphism at $$p$$, i.e. is $$F$$ a local diffeomorphism onto its image at $$p$$?

• C1. This comment, quoted here, seems to say that immersions are local diffeomorphisms onto their images. I don't know of the definition exactly for diffeomorphisms or local diffeomorphisms for non-manifolds when the non-manifold is in the range, but anyway, I wanted to try to see if immersions, whose images are submanifolds, are local diffeomorphisms onto their images. The commenter might be assuming $$N$$ has the same dimension as either $$F(N)$$ or $$M$$.

• C2. The answer here seems to say that immersions are local diffeomorphisms onto their images, but I think this is based on additional assumptions.