Nonexistence of Skew Loops on Ellipsoids

We prove that every C closed curve immersed on an ellipsoid has a pair of parallel tangent lines. This establishes the optimal regularity for a phenomenon first observed by B. Segre. Our proof uses an approximation argument with the aid of an estimate for the size of loops in the tangential spherical image of a spherical curve. A skew loop is a C closed curve without parallel tangent lines immersed in Euclidean space. B. Segre [5] was the first to prove the existence of such curves, the first explicit construction appeared in [2], and Y.-Q. Wu [9] showed examples exist in each knot class. Segre also observed that sufficiently smooth skew loops do not exist on ellipsoids [3, Appdx C]. This was an immediate consequence of a theorem of W. Fenchel [1], which states that the tangential spherical image, a.k.a. tantrix, of a (sufficiently smooth) closed spherical curve bisects the area of the sphere, when embedded. Fenchel’s theorem in turn follows quickly from the Gauss-Bonnet theorem, together with the fact that the total geodesic curvature of the tantrix of a spherical curve, when embedded, is zero [6]. But applying the Gauss-Bonnet theorem requires that the tantrix be (at least) C. Hence the original curve should be C. Other proofs of Segre’s observation [7, 8], which use Morse theory, also require C regularity. In this note we use an approximation argument to rule out the existence of skew loops on ellipsoids, without assuming extra regularity. This settles a question which had been raised in [3, Note 3.1]. The main obstacle here is that skew loops do not form an open subset in the space of loops. Indeed, small perturbations may create nearby parallel tangents. We overcome this problem by using the estimate provided in the following lemma. A mapping T : S → S is the tantrix of a C immersion γ : S ' R/2π → R provided T (t) = γ′(t)/‖γ′(t)‖. The following observation shows that the tantrix of a spherical loop may not contain small subloops: Lemma. Let T : S → S be the tantrix of a spherical curve. Suppose that there are t, s ∈ S, t 6= s, such that T (t) = T (s). Then in each of the segments of S Date: March 2003. Last Typeset August 14, 2004. 1991 Mathematics Subject Classification. Primary 53A04, 53A05; Secondary 53C45, 52A15.