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

Abstract

A smooth curve $γ: [0,1] \to \Ss^2$ 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 ${\gamma }^{\prime }(0)={\gamma }^{\prime }(1)=e_2$ 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 $(Ω\Ss^3) \vee \Ss^2 \vee \Ss^6 \vee \Ss^{10} \vee \cdots$ and $(Ω\Ss^3) \vee \Ss^4 \vee \Ss^8 \vee \Ss^{12} \vee \cdots$, respectively. As a corollary, we deduce the homotopy type of the components of the space $\Free(\Ss^1,\Ss^2)$ of free curves $γ: \Ss^1 \to \Ss^2$ (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.

arXiv comment47 pages, 13 figures

References in corpus (2)

• On the components of spaces of curves on the 2-sphere with geodesic curvature in a prescribed interval
• The geometry of the critical set of nonlinear periodic Sturm-Liouville operators

Cited by in corpus (6)

• Locally convex curves and the Bruhat stratification of the spin group
• Loose Engel structures
• Components of spaces of curves with constrained curvature on flat surfaces
• Stratification of spaces of locally convex curves by itineraries
• Characterization of some convex curves on the 3-sphere
• Homotopical and topological rigidity of hypersurfaces of spherical space forms