Homotopy Type of Spaces of Curves with Constrained Curvature on Flat Surfaces

Abstract Let S be a complete flat surface, such as the Euclidean plane. We determine the homeomorphism class 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 open interval, in terms of all parameters involved. Any connected component of such a space is either contractible or homotopy equivalent to an n -sphere, and every integer $$n\ge 1$$ n ≥ 1 is realizable. Explicit homotopy equivalences between the components and the corresponding spheres are constructed.