Foliations, solvability, and global injectivity

Let F : R n → R n be a C ∞ map such that DF ( x ) is invertible for every x ∈ R n . Although being a local diffeomorphism, F is not necessarily globally injective if n ≥ 2 . Finding additional assumptions implying the global injectivity of F for n ≥ 2 is object of intense study in several areas of mathematics. In this paper, we revisit some assumptions and relations between them in the bi-dimensional case and discuss the natural higher dimensional situation.