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Let $M$ a compact manifold, $N$ a connected manifold and $f\colon M\longrightarrow N$ an injective étale function. Why $f$ is a diffeomorphism?

I know that since $M$ is compact, $f(M)$ will be compact too, but I don't know why if $N$ is connected, I will get the diffeomorphism.

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  • $\begingroup$ What is an etale function in the smooth category? $\endgroup$ – levap Nov 30 '16 at 23:32
  • $\begingroup$ From the book where I'm reading, an étale function is one that is between manifolds of the same dimension, and is an immersion if and only if is a submersion. $\endgroup$ – MonsieurGalois Nov 30 '16 at 23:33
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An etale function is a local diffeomorphism. If $N$ is connected, $f$ is surjective: to see this remark that the image of $f$ is a closed subspace since $M$ is compact, $f(M)$ is compact. It is also open since $f$ is a local diffeomorphism. So it is a union of connected components of $N$, so it is $N$ since $N$ is connected. A local diffeomorphism which is bijective is a diffeomorphism.

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  • $\begingroup$ So if the etale function can be understood as a local diffeomorphism given the definition in the comment I added, it is not enough to say that it is a diffeomorphism? $\endgroup$ – MonsieurGalois Dec 1 '16 at 0:06
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    $\begingroup$ No, the universal cover is an etale cover, if the fundamental group is finite but it is not always a diffeomorphism $\endgroup$ – Tsemo Aristide Dec 1 '16 at 0:08
  • $\begingroup$ Thanks for the explanation. $\endgroup$ – MonsieurGalois Dec 1 '16 at 0:11

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