Controllability of the Dubins problem on surfaces

Let M be a complete oriented 2-dim Riemannian manifold. We ask the following question. Given any (p1,v1) and (p2,v2), vivelocity at pi∈ M, i = 1,2, is it possible to connect p1to p2by a curve γ with arbitrary small geodesic curvature such that, for i = 1,2,γ̇ is equal to viat pi? 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.