General helices and a theorem of Lancret

Barros, Manuel

doi:10.1090/S0002-9939-97-03692-7

General helices and a theorem of Lancret
HTML articles powered by AMS MathViewer

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