5 research outputs found
Tangent lines, inflections, and vertices of closed curves
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".Comment: Minor revisions; To appear in Duke Math.