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I have a question about immersions and smooth embeddings which i couldnt find an answer for.

I know that

  1. an immersion is locally a smooth embedding.

  2. a smooth embedding is an injective immersion which is also a topological embedding.

My question: Let's say i have an injective immersion $f:M \to N$ for every point $p$ of $M$ between two smooth manifolds $M, N$ with $\dim N\le \dim N$. Does that mean that $f$ is not only locally a smooth embedding but globally?

I'd love to get behind the relation between 1. and 2.

On a slightly more general note: If i have a function $f:M \to N$ which is a smooth embedding, what exactly does that tell me? (geometrically) For reference, whenever i have a diffeomorphism, i know that both manifolds are "basically the same shape" roughyl speaking. How do i need to think about embeddings in these regards?

Thanks you very much!

Zest
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    I explain two examples of injective immersions which fail to be embeddings in my answer here: https://math.stackexchange.com/a/1366711/68356. I think this would answer your first question at least :) – john Dec 26 '19 at 03:19
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    For your more general question: if $f$ is a smooth embedding, then its image is a submanifold of the target. And $f$ coincides with the induced inclusion map. At least, up to some self-diffeomorphism, which it doesn’t really make sense to worry about it in general. – john Dec 26 '19 at 03:23
  • Hello @john. thank you very much. Regarding my first question: what if my injective immersion is a topological embedding of $M$ into $N$? Does that mean, that my immersion is globally a smooth embedding? I'm trying to understand the relation between "local embedding" vs "global embedding" (if the latter exist) – Zest Dec 26 '19 at 03:28
  • Yes, if you have an injective immersion which is a topological embedding, then it is a smooth embedding. This should pretty much be the definition of the latter. It tells you that the difference between a local smooth embedding and a global smooth embedding is topological, which perhaps is the statement you were after? – john Dec 26 '19 at 03:32
  • i think that clarifies it! thank you very much john. happy holidays to you. – Zest Dec 26 '19 at 03:33
  • Great! Thanks, and you! – john Dec 26 '19 at 03:33

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