Spaces of curves with constrained curvature on hyperbolic surfaces

Nicolau C. Saldanha, Pedro Zühlke

Published 2016-11-28Version 1

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) $. These spaces fall into four qualitatively distinct classes, according to whether $ (\kappa_1,\kappa_2) $ contains, overlaps, is disjoint from, or contained in the interval $ [-1,1] $. Their homotopy type is computed in the latter two cases.