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

Abstract

A smooth curve γ : [0, 1] → 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 γ with γ(0) = γ(1) = e 1 and γThe space L −1,c is known to be contractible. We prove that L +1 and L −1,n are homotopy equivalent to (ΩS 3 ) ∨ S 2 ∨ S 6 ∨ S 10 ∨ · · · and (ΩS 3 ) ∨ S 4 ∨ S 8 ∨ S 12 ∨ · · · , respectively. As a corollary, we deduce the homotopy type of the components of the space Free(S 1 , S 2 ) of free curves γ : S 1 → S 2 (i.e., curves with nonzero geodesic curvature). We also determine the homotopy type of the spaces Free([0, 1], S 2 ) with fixed initial and final frames.

On the Homology of the Space of Curves Immersed in the Sphere With Curvature Constrained to a Prescribed Interval

“…In this book it is shown that the loop space ΩS Observe that L ρ 0 (Q) ⊂ I(Q). It is known from an analogous result of [22] that the inclusion map i : L ρ 0 (Q) → I(Q) induces a surjetive map on homology (refer Proposition 3.20):…”

“…A. Khesin [31] studied the topology of the space of all smooth immersed curves (not necessarily closed) with positive geodesic curvature on S 2 which start and end at given points and given directions and found the number of connected components of this space. [20], [21] and [22], N. C. Saldanha did several further works compared to Theorems 1.1 and 1.2 on the higher homotopy properties of the space of curves with positive geodesic curvature in S 2 . More precisely, he proved the following result: …”

“…Below is Lemma 6.1 of [22], the proof is based on Figure 3.4. Now we introduce a simple technical lemma:…”

“…The following results are adapted directly from [22] of N. C. Saldanha. The following result corresponds to the Lemma 6.2 in this article.…”

“…The following lemma, which is a direct adaptation of Lemma 6.6 in [22], guarantees the continuity of the choice on Step 1: …”