When is a vector field injective?

In this paper we define a notion of injectivity for a vector field over a Riemannian manifold and we give a sufficient condition for it. The proof is an extension of a well known Theorem stating that a map from an open convex set of the euclidean space into is a diffeomorphism onto its image provided that the bilinear operator associated to its differential is (positive or negative) definite everywhere. In the second part of the paper, we show that these results have a natural extension to the tangent bundle, with straightforward applications to second order differential equations.