Global Rank Theorem

Let \(F\colon M\to N\) be a smooth map between smooth manifolds with constant rank .

1. If \(F\) is surjective, then it is a smooth submersion
2. If \(F\) is injective, then it is a smooth immersion
3. If \(F\) is bijective, then it is a diffeomorphism

[@lee2013smooth_manifolds, Theorem 4.14]