Closed curves of constant torsion. II
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- by Joel L. Weiner PDF
- Proc. Amer. Math. Soc. 67 (1977), 306-308 Request permission
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
- Werner Fenchel, On the differential geometry of closed space curves, Bull. Amer. Math. Soc. 57 (1951), 44–54. MR 40040, DOI 10.1090/S0002-9904-1951-09440-9
- Joel L. Weiner, Closed curves of constant torsion, Arch. Math. (Basel) 25 (1974), 313–317. MR 346712, DOI 10.1007/BF01238680
Additional Information
- © Copyright 1977 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 67 (1977), 306-308
- MSC: Primary 53C40
- DOI: https://doi.org/10.1090/S0002-9939-1977-0461385-0
- MathSciNet review: 0461385