The homotopy type of spaces of locally convex curves in the sphere

A smooth curve \gamma: [0,1] \to \Ss^2 is locally convex if its geodesic curvature is positive at every point. J. A. Little showed that the space of all locally convex curves $\gamma$ with $\gamma(0) = \gamma(1) = e_1$ and $\gamma'(0) = \gamma'(1) = e_2$ has three connected components $L_{-1,c}$, $L_{+1}$, $L_{-1,n}$. The space \cL_{-1,c} is known to be contractible. We prove that \cL_{+1} and \cL_{-1,n} are homotopy equivalent to (\Omega\Ss^3) \vee \Ss^2 \vee \Ss^6 \vee \Ss^{10} \vee \cdots and (\Omega\Ss^3) \vee \Ss^4 \vee \Ss^8 \vee \Ss^{12} \vee \cdots, respectively. As a corollary, we deduce the homotopy type of the components of the space \Free(\Ss^1,\Ss^2) of free curves \gamma: \Ss^1 \to \Ss^2 (i.e., curves with nonzero geodesic curvature). We also determine the homotopy type of the spaces \Free([0,1], \Ss^2) with fixed initial and final frames.Comment: 47 pages, 13 figure

Homotopy type of spaces of curves with constrained curvature on flat surfaces

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 $n\geq 1$ is realizable. Explicit homotopy equivalences between the components and the corresponding spheres are constructed.Comment: 39 pages, 13 figures. Differs from previous version by many improvements of the expositio