Integral Curvatures, Oscillation and Rotation of Spatial Curves around Affine Subspaces

The main result of the paper is an upper bound for oscillation of spatial curves around geodesic subspaces of the ambient space in terms of the integral geodesic curvatures of the curves. Let Mn be the Euclidean space Rn, projective space Pn or the sphere Sn equipped with the Riemannian metric of Gaussian curvature c(M) = 0, 1 or r−n > 0 respectively, and Γ ⊂ M be a smooth curve parameterized by the arc length s ∈ [0, `]. For such curves the (geodesic) Frenet curvatures κ1(s), . . . ,κn−1(s) can be defined, the last one up to the choice of sign in the non-orientable case of Pn. The generalized inflection points are defined by the condition that the last curvature κn−1(s) vanishes. We prove that the number of intersections of Γ with an arbitrary affine hyperplane Ln−1 ⊂ Rn (respectively, any equator of codimension 1 in the sphere or a projective hyperplane in Pn) can be at most 1/π times the sum w0Kn(Γ )+w1Kn−1(Γ )+· · ·+wn−1K1(Γ )+wnK0(Γ )+wn+1K−1(Γ ), where: K1(Γ ), . . . ,Kn−1(Γ ) are (absolute) integral Frenet curvatures of Γ , Kn(Γ ) = π × (number of generalized inflection points), K0(Γ ) = c1/n(M) · |Γ |, where |Γ | is the Riemannian length of Γ , K−1(Γ ) = 0 or π is π/2 times the number of endpoints of Γ , w0 = w1 = 1, w2 = 2, wj = j − 1 for j > 3 are the universal weights. For curves in the Euclidean case M = Rn a similar estimate can be found for properly defined rotation around affine subspaces of arbitrary dimension k between 0 and n−2. We show that this rotation can be at most w0Kk+1(Γ )+ · · · + wk−1K1(Γ ) + wk+1K−1(Γ ), where the term wkK0(Γ ) is missing since c(Rn) = 0. The proof is based on arguments from integral geometry (alias geometric probability) and non-oscillation theory for ordinary linear equations. 1. Oscillation and rotation around affine subspaces The principal question addressed in this paper, can be formulated as follows: given a smooth curve Γ in the Euclidean space R with known integral Frenet curvatures ∫ Γ |κj(s)| ds, j = 1, . . . , n − 1, give an upper bound for the number of intersections of Γ with any affine hyperplane A ⊂ R, dimA = n − 1, and, more Date: Submitted December 15, 1995. 1991 Mathematics Subject Classification. Primary 53A04, 34A26, Secondary 34A30, 34C10, 53C65.