  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.
