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 $S{O}_3$ ($n>=3$). 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 $C_{k_1^{\prime }}^{k_2^{\prime }}$ whenever $r_1-r_2=r_1^{\prime }-r_2^{\prime }$ ($r_i^{\prime }=arccot{k}_i^{\prime }$) are also presented.

117 pages, 20 figures. This is the unrevised version of my Ph.D. thesis. It is stored here only because it contains a few proofs that were omitted in an article which was based on the thesis. Those who are interested in the subject should read this article instead, which was written later and has a more polished exposition. It can be found at arXiv:1304.2629

Cited by in corpus (1)

• Topology of the space of locally convex curves on the 3-sphere