# $\tilde F$ immersion is equivalent to $F$ immersion

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?

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.