General helices and a theorem of Lancret
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- by Manuel Barros PDF
- Proc. Amer. Math. Soc. 125 (1997), 1503-1509 Request permission
Abstract:
We present a theorem of Lancret for general helices in a 3-dimen- sional real-space-form which gives a relevant difference between hyperbolic and spherical geometries. Then we study two classical problems for general helices in the 3-sphere: the problem of solving natural equations and the closed curve problem.References
- M.Barros, A.Ferrández, P.Lucas and M.A.Meroño, Helicoidal filaments in the 3-sphere. Preprint.
- N.V.Efimov, Nekotorye zadachi iz teorii prostranstvennykh krivykh. Usp.Mat.Nauk, 2 (1947), 193-194.
- Charles Hopkins, Rings with minimal condition for left ideals, Ann. of Math. (2) 40 (1939), 712–730. MR 12, DOI 10.2307/1968951
- M.A.Lancret, Mémoire sur les courbes à double courbure. Mémoires présentés à l’Institut 1 (1806), 416-454.
- Joel Langer and Ron Perline, Local geometric invariants of integrable evolution equations, J. Math. Phys. 35 (1994), no. 4, 1732–1737. MR 1267918, DOI 10.1063/1.530567
- Joel Langer and David A. Singer, The total squared curvature of closed curves, J. Differential Geom. 20 (1984), no. 1, 1–22. MR 772124
- Joel Langer and David A. Singer, Knotted elastic curves in $\textbf {R}^3$, J. London Math. Soc. (2) 30 (1984), no. 3, 512–520. MR 810960, DOI 10.1112/jlms/s2-30.3.512
- Richard S. Millman and George D. Parker, Elements of differential geometry, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1977. MR 0442832
- Barrett O’Neill, Semi-Riemannian geometry, Pure and Applied Mathematics, vol. 103, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. With applications to relativity. MR 719023
- U. Pinkall, Hopf tori in $S^3$, Invent. Math. 81 (1985), no. 2, 379–386. MR 799274, DOI 10.1007/BF01389060
- Paul D. Scofield, Curves of constant precession, Amer. Math. Monthly 102 (1995), no. 6, 531–537. MR 1336639, DOI 10.2307/2974768
- Dirk J. Struik, Lectures on classical differential geometry, 2nd ed., Dover Publications, Inc., New York, 1988. MR 939369
Additional Information
- Manuel Barros
- Affiliation: Departamento de Geometria y Topologia, Facultad de Ciencias, Universidade de Granada, 18071 Granada, Spain
- Email: mbarros@goliat.ugr.es
- Received by editor(s): July 26, 1995
- Received by editor(s) in revised form: November 14, 1995
- Additional Notes: Partially supported by DGICYT Grant No. PB94-0750.
- Communicated by: Christopher Croke
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 1503-1509
- MSC (1991): Primary 53C40, 53A05
- DOI: https://doi.org/10.1090/S0002-9939-97-03692-7
- MathSciNet review: 1363411