Let F: R^n -> R^n be a C^\infty map such that DF(x) is invertible for all x
in R^n. We know that F is a local diffeomorphism but, in general, it is not
globally injective. We discuss the relations between some additional hypothesis
that guarantee the global injectivity of F. Further, based on one of these
hypotheses, we establish a necessary condition for the existence of F: R^n ->
R^n such that det DF = h, where h: R^n -> [0,\infty) is a given C^\infty
function