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Given smooth manifold $M$ and it's submanifold $S$(e.g. open subset of $M$) we have inclusion map $i:S\to M$.

And we treat $i$ as $i(x) = x$ typically.

For example $i:S^n \to \mathbb{R}^{n+1}$ is valid to define $i(x) = x$ But it seems not for example inclusion $i:\mathbb{R}^n \to \mathbb{R}^{n+1}$ as $(x_1,...,x_n) \to (x_1,...,x_n,0)$

So I was a bit confused what is the definition for inclusion here?Should we treat it as $i(x) = x$?

Is this "inclusion" a topological embedding by default setting or not?

I found an explanation here

yi li
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2 Answers2

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(1)the inclusion map is always defined as $i(x) = x$ for $i:S\to M$ such that $S\subset M$.

(2)Sometimes we may call inclusion just as a injection map,for example $i:\Bbb{R}^n\to \Bbb{R}^{n+1}$ defined above are injection.

note that both definition of "inclusion" need only set level things,no topology or smooth structure.

(For immersed/embedded submanifold defintion inclusion is the first meaning.

we call this "inclusion" the "immersion/embedding" then since there exist some additional structure on it.)

yi li
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  • wait in the last line do you really mean 'as topological imbedding' instead of, like, 'as immersion' ? – BCLC Apr 26 '21 at 03:44
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    @BCLC If topology on immersed submanifold happends to be the subspace topology,we can see $i:S\to M$ is homeomorphic onto it's image ,so it's a topological embedding – yi li Apr 26 '21 at 04:24
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    We know topology on immersed submanifold needs not to be subspace topology – yi li Apr 26 '21 at 04:27
  • aaaaaahhhhhhhhhh wait i realise my mistake. i thought $S$ is an immersed submanifold if and only if $i: S \to M$ is an immersion, but '$i: S \to M$ is an immersion' is nonsensical because saying a map is an immersion assumes both domain and range are smooth manifolds. – BCLC Apr 26 '21 at 04:40
  • right thanks anyway never mind lol – BCLC Apr 26 '21 at 04:40
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    Your second paragraph is wrong... $\mathbb R^n$ is not by definition a subset of $\mathbb R^{n+1}$. The "inclusion" here is really an injective map $\mathbb R^n \to \mathbb R^{n+1}$, and is not the inclusion in the ordinary sense. – Arctic Char Apr 26 '21 at 04:44
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    @Arctic Char you are right,this is post long time ago. – yi li Apr 26 '21 at 04:45
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There are actually deeper subtleties: The notion of a submanifold can breed much confusion: Do you want a submanifold to be immersed, do you want it to be an embedded submanifold?

An immersed submanifold $S$ of a manifold of $M$ is the image of a manifold under an immersion. An immersion is a smooth map with injective derivative.

An embedding is a topological embedding, i.e., a homeomorphism onto its image (with respect to the subspace topology), that is also an injective immersion.

Note!: Immersions are not necessarily injective, nor a topological embedding!

AmorFati
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  • I know what you mean,I want to know what's the inclusion mean? Should we prove it's topological embedding first if the map is called "inclusion"? If we want to prove the map is smooth embedding? I know for general map ,we need first show it's topological embedding ,what about inclusion map? – yi li Aug 20 '20 at 09:17
  • @yi_li See https://math.stackexchange.com/questions/707010/is-the-inclusion-map-always-smooth Does this help? – AmorFati Aug 20 '20 at 09:20
  • Hi,thanks,I know what you are saying now. By the way,the inclusion map is not necessarily topological embedding in manifold setting correct? – yi li Aug 24 '20 at 07:50
  • Relevant: When is an inclusion map smooth? maybe ? It seems part of OP's confusion is the idea of inclusion like $\iota(x)=(x,0)$ vs $i(x)=x$, for $\iota: \mathbb R^n \to \mathbb R^{n+1}$ but $i: \mathbb R^{n+1} \to \mathbb R^{n+1}$ – BCLC Apr 26 '21 at 03:42