Let $M$ be a complete, connected, two-dimensional Riemannian manifold. Consider the following question: Given any $(p_1,v_1)$ and $(p_2,v_2)$ in $TM$, is it possible to connect $p_1$ to $p_2$ by a curve $\gamma$ in $M$ with arbitrary small geodesic curvature such that, for $i=1,2$, $\dot \gamma$ is equal to $v_i$ at $p_i$? In this paper, we bring a positive answer to the question 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.
