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Related: Equivalent definitions for smooth embedding?, Are manifold subsets submanifolds?

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

Let $N$ and $M$ be smooth manifolds of respective dimensions $n$ and $m$. Let $F:N \to M$ be a smooth map. Let $F(N)$ be a smooth (regular/embedded) submanifold of $M$ and not simply a manifold subset. Let $i: F(N) \to M$ be inclusion. Let $\tilde F$ be $F$ with restricted range onto its image, i.e. $\tilde F: N \to F(N)$, $F = i \circ \tilde F$. I think the statements below are true.

C. $\tilde F$ is an immersion $\Rightarrow$ $F$ is an immersion

D. $\tilde F$ is an immersion $\Leftarrow$ $F$ is an immersion

Question 1. Are these proofs correct?

For both C and D:

  • CD0. Weaker version of Theorem 11.14: Inclusion maps from submanifolds are smooth. (I'm actually not sure if Theorem 11.14 proves this. I ask about this here.)

  • CD1. $\tilde F$ in the first place is smooth because $F(N)$ is a submanifold and because of (need not be only because of. I might ask about this in another question) Theorem 11.15. Therefore, ${\tilde F}_{*,p}$ is defined for each $p \in N$ and thus it makes sense to talk about whether or not $\tilde F$ is an immersion, submersion, etc

  • CD2. Similarly, $i$ is smooth because (need not be only because. I might ask about this in another question) $F(N)$ is a submanifold and because of (CD0). Therefore, $i_{*,F(p)}$ is defined for each $F(p) \in F(N)$, and thus it makes sense to discuss whether or not $i$ is an immersion, a submersion, etc.

For C:

  • C2. Weaker version of Theorem 11.14: Inclusion maps from submanifolds are (not only smooth but also) immersions (also they are embeddings).

  • C3. Because $F(N)$ is a submanifold, $i$ is an immersion by (C2) and (CD2).

  • C4. By Chain Rule, (CD1) and (CD2), $F_{*,p} = (i \circ \tilde F)_{*,p} = i_{*,F(p)} \circ \tilde F_{*,p}$

  • C5. Therefore, $F_{*,p}$ is injective, by (C4) and by injectivity of ${\tilde F}_{*,p}$ because compositions of injections are injective

For D:

  • D2. By Chain Rule, (CD1) and (CD2), $F_{*,p} = (i \circ \tilde F)_{*,p} = i_{*,F(p)} \circ \tilde F_{*,p}$

  • D3. $g \circ f$ injective implies $f$ injective, so I think (D2) and injectivity of $F_{*,p}$ give us injectivity of $\tilde F_{*,p}$. (see Question 2)

Question 2. For (D3), I think we don't care about whether or not $i$ is an immersion. Is this correct?

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The proofs are correct. Exactly in the same way you can also show the slightly stronger statement:

Let $F:M\to N$ a smooth map between smoth manifolds und $W$ an embedded submanifold of $N$ such that $f(M)\subseteq W$. Then $F$ is an immersion iff the restriction $\tilde F:M\to W$ is an immersion.

Also your remark concerning (D3) is correct. To say we don't care if $i$ is an immersion means that we can generalize to:

Let $F:M\to N $ and $G:N\to Q$ be smooth maps between smooth manifolds. If $G\circ F$ is an immersion then $F$ is an immersion.

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  • $\begingroup$ Thanks! Btw, I assume all your manifolds have dimension. They could be true if they don't have (uniform) dimensions, but I choose to not think of that now. $\endgroup$ – user636532 Jul 22 '19 at 5:35

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