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

الأصل · EN

A smooth curve γ: [0,1] → 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 γ with γ(0) = γ(1) = e₁ and γ'(0) = γ'(1) = e₂ has three connected components L₋₁,c, L₊₁, L₋₁,ₙ. The space ₋₁,c is known to be contractible. We prove that ₊₁ and ₋₁,ₙ are homotopy equivalent to (Ω³) ² ⁶ ¹⁰ and (Ω³) ⁴ ⁸ ¹², respectively. As a corollary, we deduce the homotopy type of the components of the space (¹,²) of free curves γ: ¹ → ² (i.e., curves with nonzero geodesic curvature). We also determine the homotopy type of the spaces ([0,1], ²) with fixed initial and final frames.