Spaces of Locally Convex Curves in S^n and Combinatorics of the Group B_(n+1)^+

Abstract

Abstract. In the 1920's Marston Morse developed what is now known asMorse theory trying to study the topology of the space of closed curves on S 2 ([7], [5]). We propose to attack a very similar problem, which 80 years later remains open, about the topology of the space of closed curves on S 2 which are locally convex (i.e., without inflection points). One of the main difficulties is the absence of the covering homotopy principle for the map sending a nonclosed locally convex curve to the Frenet frame at its endpoint.In the present paper we study the spaces of locally convex curves in S n with a given initial and final Frenet frames. Using combinatorics of B + n+1 = B n+1 ∩ SO n+1 , where B n+1 ⊂ O n+1 is the usual Coxeter-Weyl group, we show that for any n ≥ 2 these spaces fall in at most n 2 + 1 equivalence classes up to homeomorphism. We also study this classification in the double cover Spin(n + 1). For n = 2 our results complete the classification of the corresponding spaces into two topologically distinct classes, or three classes in the spin case.

Untitled

“…To say that the latter does not vanish is equivalent to requiring that the first n (covariant) derivatives of γ at γ (t) span the tangent space to M at this point for each t ∈ [0, 1]. Some papers treating this problem, especially for spaces of closed curves on the simplest manifolds, such as ‫ޒ‬ n , ‫ޓ‬ n or ‫ސޒ‬ n , include [Anisov 1998;Feldman 1968;1971, Khesin andShapiro 1992;, Little 1970Mostovoy and Sadykov 2012;Saldanha and Shapiro 2012;Shapiro 1991, Shapiro 1993]. Most of these are concerned with obtaining characterizations of the connected components of the corresponding spaces.…”

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

“…Indeed there are some studies: [24], [32], [23], [1], [2] and [14]. The research on the topological aspects of spaces of curves has not been restricted exclusively to sphere S n .…”