Let M be a complete oriented 2-dim Riemannian manifold. We ask the following question. Given any (pi, vi) and (p(2), v(2)), v(i) velocity at p(i) is an element of M, i = 1, 2, is it possible to connect pi to p(2) by a curve gamma with arbitrary small geodesic curvature such that, for i = 1, 2, gamma is equal to v(i) at p(i)? In this paper, we prove that the answer to the question is positive if M verifies one of the following three conditions (a) M is compact, (b) M is asymptotically flat, (c) M has bounded non negative curvature outside a compact subset.