Show that an embedding from a compact, smooth manifold to a connected smooth manifold is a diffeomorphism.

Question

Let $M,N$ be nonempty, smooth manifold of the same dimension. Let $N$ be compact and $M$ be connected. Suppose that $f:N\rightarrow M$ is an embedding. We want to show that $f$ is a diffeomorphism.

Since $f$ is an embedding, then we know that

$f(N)\subset M$ is a smooth submanifold, and
$f:N\rightarrow f(N)$ is a diffeomorphism. That is we identify the domain $N$ with its image $f(N)\subset M$ such that $N\subset M$.

We would like to show that $f$ is a diffeomorphism. So, we need to show $f$ is an invertible smooth map.

In the first paragraph you mean $f : N \rightarrow M$ ? – Kelvin Lois Nov 29 '17 at 18:49 — Kelvin Lois, Nov 29 '17 at 18:49
Yes, I mean $f:N\rightarrow M$ – user394412 Nov 29 '17 at 18:52 — user394412, Nov 29 '17 at 18:52

score 0 · Answer 1 · answered Nov 29 '17 at 20:55

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I found the answer here: Smooth embeddings that are homeomorphisms but not diffeomorphisms

Since we have an embedding, then $f$ is an immersion.

Since $f$ is an immersion and the dimension of $M$ and $N$ are the same, then we have a local diffeomorphism.

Therefore, $f^{-1}:M\rightarrow N$ is also smooth. We have shown $f$ is an invertible smooth map. Hence, $f$ is a diffeomorphism.

answered Nov 29 '17 at 20:55

user394412

169

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You also need to show that $f$ is surjective. – Jack Lee Nov 29 '17 at 23:07
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How would I show that? – user394412 Nov 30 '17 at 18:16
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Hint: Connectedness of $M$ is crucial. Do you know the typical argument for showing some subset of a connected space is the whole space? – Jack Lee Nov 30 '17 at 18:20
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No I do not know this. – user394412 Nov 30 '17 at 18:27
@JackLee Sounds familiar. what is it please?is it to do with connected components? – Jun 20 '19 at 14:39
@JackLee Oh wait I think I know. We can't have $S$ to be proper, open and closed in connected nonempty $C$, so prove $S$ is nonempty open and closed in $C$ to deduce $S=C$? – Jun 20 '19 at 14:48
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@SeleneAuckland: Yes, that's the argument I was referring to. – Jack Lee Jun 20 '19 at 19:36
Thanks Prof @JackLee! – Jun 21 '19 at 06:48

Show that an embedding from a compact, smooth manifold to a connected smooth manifold is a diffeomorphism.

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