Homotopies of Curves on the 2-Sphere with Geodesic Curvature in a Prescribed Interval

Abstract

Let $C_{k_1}^{k_2}$ denote the set of all closed curves of class $C^r$ on the sphere $S^2$ whose geodesic curvatures are restricted to lie in $(k_1,k_2)$, furnished with the $C^r$ topology (for some $r >= 2$ and possibly infinite $k_1 < k_2$). In 1970, J. Little proved that the space $C_0^{+\infty}$ of closed curves having positive geodesic curvature has three connected components. Let $r_i = arccot k_i$ (i = 1, 2). We show that $C_{k_1}^{k_2}$ has n connected components $C_1, ..., C_n$, where n is the greatest integer smaller than or equal to $\pi/(r_1-r_2) + 1$, and $C_j$ contains circles traversed j times ($1 <= j <= n$). The component $C_{n-1}$ also contains circles traversed $(n-1) + 2m$ times, and $C_n$ also contains circles traversed $n + 2m$ times, for any natural number m. In addition, each of $C_1, ..., C_{n-2}$ is homotopy equivalent to $SO_3$ ($n >= 3$). A simple characterization of the components in terms of the properties of a curve and a proof that $C_{k_1}^{k_2}$ is homeomorphic to $C_{k'_1}^{k'_2}$ whenever $r_1 - r_2 = r'_1 - r'_2$ ($r'_i = arccot k'_i$) are also presented.