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Proceedings of the American Mathematical Society

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Closed curves of constant torsion. II

Author: Joel L. Weiner
Journal: Proc. Amer. Math. Soc. 67 (1977), 306-308
MSC: Primary 53C40
MathSciNet review: 0461385
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Abstract: In this note we show that there exist closed regular $ {C^3}$ space curves $ \alpha $ with curvature $ \kappa > 0$ and nonzero constant torsion $ \tau $ whose total torsion $ \smallint_\alpha {\tau \;ds} $ is arbitrarily small. In so doing, we give another proof of the existence of closed curves of nonzero constant torsion. This note shows that Conjecture 2 in [2] is incorrect since the preceding statement is equivalent to the statement that there exist closed curves of constant torsion $ \tau = 1$ whose length is arbitrarily small.

References [Enhancements On Off] (What's this?)

  • [1] Werner Fenchel, On the differential geometry of closed space curves, Bull. Amer. Math. Soc. 57 (1951), 44-54. MR 12, 634. MR 0040040 (12:634d)
  • [2] Joel L. Weiner, Closed curves of constant torsion, Arch. Math. (Basel) 25 (1974), 313-317. MR 49 #11437. MR 0346712 (49:11437)

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Keywords: Closed space curves, constant torsion, total torsion, binormal indicatrix, Peano direction
Article copyright: © Copyright 1977 American Mathematical Society

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