Spaces of curves with constrained curvature on hyperbolic surfaces

Let $S$ be a hyperbolic surface. We investigate the topology of the space of all curves on $S$ which start and end at given points in given directions, and whose curvatures are constrained to lie in a given interval $(\kappa_1,\kappa_2)$. Such a space falls into one of four qualitatively distinct classes, according to whether $(\kappa_1,\kappa_2)$ contains, overlaps, is disjoint from, or contained in the interval $[-1,1]$. Its homotopy type is computed in the latter two cases. We also study the behavior of these spaces under covering maps when $S$ is arbitrary (not necessarily hyperbolic nor orientable) and show that if $S$ is compact then they are always nonempty.