Spaces of Curves with Constrained Curvature on Hyperbolic Surfaces

摘要

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.