ON THE COMPONENTS OF SPACES OF CURVES ON THE 2-SPHERE WITH GEODESIC CURVATURE IN A PRESCRIBED INTERVAL

Abstract

Let [Formula: see text] denote the set of all closed curves of class Cron the sphere S2whose geodesic curvatures are constrained to lie in (κ1, κ2), furnished with the Crtopology (for some r ≥ 2 and possibly infinite κ1< κ2). In 1970, J. Li ttle proved that the space [Formula: see text] of closed curves having positive geodesic curvature has three connected components. Let ρi= arccot κi(i = 1, 2). We show that [Formula: see text] has n connected components [Formula: see text] where [Formula: see text] and [Formula: see text] contains circles traversed j times (1 ≤ j ≤ n). The component [Formula: see text] also contains circles traversed (n - 1) + 2k times, and [Formula: see text] also contains circles traversed n + 2k times, for any k ∈ N. Further, each of [Formula: see text](n ≥ 3) is homeomorphic to SO3× E, where E is the separable Hilbert space. We also obtain a simple characterization of the components in terms of the properties of a curve and prove that [Formula: see text] is homeomorphic to [Formula: see text] whenever [Formula: see text].