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

摘要

Let C_(κ1)~(κ2) denote the set of all closed curves of class C~r on the sphere S~2 whose geodesic curvatures are constrained to lie in (κ1, κ2), furnished with the C~r topology (for some r ≥ 2 and possibly infinite κ1 < κ2). In 1970, J. Little proved that the space C_0~(+∞) of closed curves having positive geodesic curvature has three connected components. Let ρ_i = arccot κ_i (i = 1, 2). We show that C_(κ1)~(κ2) has n connected components C_1, . . ., C_n, where n =[π/ρ_1 ? ρ_2]+ 1 and C_j contains circles traversed j times (1 ≤ j ≤ n). The component C_(n?1) also contains circles traversed (n?1)+2k times, and Cn also contains circles traversed n+2k times, for any k ∈ N. Further, each of C_1, . . .,C_(n?2) (n ≥ 3) is homeomorphic to SO_3 × 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 C_(κ1)~(κ2) is homeomorphic to C_(κ1)~(κ2) whenever ρ_1 ? ρ_2 =ρ_1 ?ρ_2 (ρ_i = arccot κ_i).