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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:

  1. What can we say about immersions whose images are actually submanifolds? Or surjective immersions?

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

  3. Immersion and Local embedding


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.

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