Dubins ’ Problem on Surfaces . I . Nonnegative Curvature

Let M be a complete, connected, two-dimensional Riemannian manifold. Consider the following question: Given any (p1, v1) and (p2, v2) in T M , is it possible to connect p1 to p2 by a curve γ in M with arbitrary small geodesic curvature such that, for i = 1, 2, γ̇ is equal to vi at pi? In this article, 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, and (c) M has bounded nonnegative curvature outside a compact subset.