Tangent lines, inflections, and vertices of closed curves

Abstract

Abstract. We show that every smooth closed curve Γ immersed in Euclidean space R 3 satisfies the sharp inequality 2(P + I) + V ≥ 6 which relates the numbers P of pairs of parallel tangent lines, I of inflections (or points of vanishing curvature), and V of vertices (or points of vanishing torsion) of Γ. We also show that 2(P + + I) + V ≥ 4, where P + is the number of pairs of concordant parallel tangent lines. The proofs, which employ curve shortening flow with surgery, are based on corresponding inequalities for the numbers of double points, singularities, and inflections of closed curves in the real projective plane RP 2 and the sphere S 2 which intersect every closed geodesic. These findings extend some classical results in curve theory including works of Möbius, Fenchel, and Segre, which is also known as Arnold's "tennis ball theorem".

Boundary torsion and convex caps of locally convex surfaces

“…In this sense, Theorem 1.1 is a substantial extension of the classical four vertex theorem for planar curves. Indeed points of vanishing torsion of a space curve are natural generalizations of critical points of curvature of a planar curve, e.g., see [23,Note 1.5]. We should also note that the local nonflatness assumption along ∂M here is necessary to ensure that the sign of τ behaves in the claimed manner, see Example 7.1.…”

“…The study of special points of curvature and torsion of closed curves has generated a vast and multifaceted literature since the works of Mukhopadhyaya [44] and A. Kneser [39] on vertices of planar curves were published in 1910-1912, although aspects of these investigations may be traced even further back to the study of inflections by Möbius [42] and Klein [38], see [24,23]. The first version of the four vertex theorem for space curves, which was concerned with curves lying on smooth strictly convex surfaces, was stated by Mohrmann [43] in 1917, and proved by Barner and Flohr [9] in 1958.…”

“…Among various applications of four vertex theorems, we mention a paper of Berger and Calabi et al [52] on physics of floating bodies, and recent work of Bray and Jauregui [14] in general relativity. See also the works of Arnold [7,8] for relations with contact geometry, the book of Ovsienko and Tabachnikov [47] for projective geometric aspects, Angenent [6] for connections with mean curvature flow, which are also discussed in [23], and Ivanisvili et al [36,35] for applications to the study of Bellman functions. Other references and more background on four vertex theorems may be found in [18,65,22,48,26].…”

On curves with nonnegative torsion

“…The four-vertex theorem of Sedykh [10] for closed convex curves γ in R 3 asserts that the torsion of γ has at least four zeroes. A refinement of this result due to Thorbergsson and Umehara [11] shows that the torsion must change sign at least four times (if γ is not a plane curve), which implies Theorem 1 (see also [6]). Theorem 2 follows as well because the sign of the torsion is determined by the sign of the expression, (γ × γ ) · γ , which is independent of whether × and · are taken with respect to the R 3 or R 2,1 metric.…”

“…To prove existence, we write down the formula for f which comes from identifying the relevant kernel and verify that this f satisfies equation (6). For convenience, let p = 1/κ 0 , and recall that κ 0 , p, and f are periodic functions with period 2π.…”