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

Abstract

A smooth curve $\gamma :\left[0,1\right]\to {\mathbb{S}}^2$ 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 $\gamma \left(0\right)=\gamma \left(1\right)=e_1$ and ${\gamma }^{\prime }\left(0\right)={\gamma }^{\prime }\left(1\right)=e_2$ has three connected components ${\mathcal{ℒ}}_{-1,c}$, ${\mathcal{ℒ}}_{+1}$, ${\mathcal{ℒ}}_{-1,n}$. The space ${\mathcal{ℒ}}_{-1,c}$ is known to be contractible. We prove that ${\mathcal{ℒ}}_{+1}$ and ${\mathcal{ℒ}}_{-1,n}$ are homotopy equivalent to $\left(\Omega {\mathbb{S}}^3\right)\vee {\mathbb{S}}^2\vee {\mathbb{S}}^6\vee {\mathbb{S}}^{10}\vee \cdots$ and $\left(\Omega {\mathbb{S}}^3\right)\vee {\mathbb{S}}^4\vee {\mathbb{S}}^8\vee {\mathbb{S}}^{12}\vee \cdots$, respectively. As a corollary, we deduce the homotopy type of the components of the space $Free\left({\mathbb{S}}^1,{\mathbb{S}}^2\right)$ of free curves $\gamma :{\mathbb{S}}^1\to {\mathbb{S}}^2$ (ie curves with nonzero geodesic curvature). We also determine the homotopy type of the spaces $Free\left(\left[0,1\right],{\mathbb{S}}^2\right)$ with fixed initial and final frames.