Central Cross-Sections Make Surfaces of Revolution Quadric

Solomon, Bruce

doi:10.4169/193009709x470209

Cited by 3 publications

(1 citation statement)

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“…This observation is meaningful because there exist closed space curves, called skew loops, without any pairs of parallel tangent lines. Skew loops, which were first studied by Segre [41], may be constructed in each knot class [51], and have been the subject of several recent works [15,18,16,21,20,43] due in part to their interesting connections with quadric surfaces. We should also mention that (2) had been observed by Segre [41] in the special case where P + = I = 0, while (1) appears to be entirely new.…”

Section: Introductionmentioning

confidence: 99%

Tangent lines, inflections, and vertices of closed curves

Ghomi¹

2013

Duke Math. J.

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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".

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Section: Introductionmentioning

confidence: 99%

Tangent lines, inflections, and vertices of closed curves

Ghomi¹

2013

Duke Math. J.

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Central figure-8 cross-cuts make surfaces cylindrical

Solomon

2017

Advances in Geometry

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We prove: If a complete connected C 2 surface M in R 3 has general position, intersects some plane along a clean figure-8 (a loop with total curvature zero) and all compact intersections with planes have central symmetry, then M is a (geometric) cylinder over some central figure-8. On the way, we establish interesting facts about centrally symmetric loops in the plane; for instance, a clean loop with even rotation number 2k can never be central unless it passes through its center exactly twice and k = 0 . §1 §2 §3Ref.

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Surfaces with central convex cross-sections

Solomon¹

2012

Comment. Math. Helv.

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Abstract. Say that a surface in S ⊂ R 3 has the central plane oval property, or cpo, if• S meets some affine plane transversally along an oval, and • Every such transverse plane oval on S has central symmetry. We show that a complete, connected C 2 surface with cpo must either be a cylinder over a central oval, or else quadric. We apply this to deduce that a complete C 2 surface containing a transverse plane oval but no skewloop, must be cylindrical or quadric.

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Central Cross-Sections Make Surfaces of Revolution Quadric

Cited by 3 publications

References 6 publications

Tangent lines, inflections, and vertices of closed curves

Tangent lines, inflections, and vertices of closed curves

Central figure-8 cross-cuts make surfaces cylindrical

Surfaces with central convex cross-sections

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