Tangent lines, inflections, and vertices of closed curves

Details Tangent lines, inflections, and vertices of closed curves Tangent lines, inflections, and vertices of closed curves

2012-01-06

arxiv.org/abs/1201.1515v3

Mathematics

Differential Geometry

32 pages and 12 figures; Typesetting error in references corrected

Scientific paper

We show that every smooth closed curve C immersed in Euclidean 3-space satisfies the sharp inequality 2(P+I)+V >5 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 C. We also show that 2(P'+I)+V >3, 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 and the sphere which intersect every closed geodesic. These findings extend some classical results in curve theory including works of Moebius, Fenchel, and Segre, which is also known as Arnold's "tennis ball theorem".

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