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.
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.