General helices and a theorem of Lancret. (English) Zbl 0876.53035

In this paper, the author defines the concept of general helix in a 3-dimensional real space form \(M\) and presents a theorem of Lancret for such a class of curves. He studies two classical problems for general helices: the problem of solving natural equations and the closed curve problem, which is an open problem in elementary differential geometry. It should be pointed out that the main results give relevant differences between hyperbolic and spherical geometries: while the spherical case is nicely similar to the original Lancret theorem, in the hyperbolic version there are no nontrivial general helices. As for the closed curve problem, the author shows that in the 3-sphere there exists a rational one-parameter family of closed general helices living in the Hopf torus shaped on any immersed closed curve in the 2-sphere. These curves arise, as the author indicates, in the context of the interplay between geometry and integrable Hamiltonian systems. Therefore there is a natural geometric evolution on general helices inducing a mKdV (modified Kortewed-de Vries) curvature evolution equation coming from the LIE (localized induction equation). The role of general helices here is probably similar to that of curves of constant torsion or constant natural curvature. The paper is nicely written and easy to read.


53C40 Global submanifolds
53A04 Curves in Euclidean and related spaces
53A05 Surfaces in Euclidean and related spaces
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[1] M.Barros, A.Ferrández, P.Lucas and M.A.Meroño, Helicoidal filaments in the 3-sphere. Preprint.
[2] N.V.Efimov, Nekotorye zadachi iz teorii prostranstvennykh krivykh. Usp.Mat.Nauk, 2 (1947), 193-194.
[3] W.Fenchel, The differential geometry of closed space curves. Bull.Amer.Math.Soc., 57 (1951), 44-54. · Zbl 0042.40006
[4] M.A.Lancret, Mémoire sur les courbes à double courbure. Mémoires présentés à l’Institut 1 (1806), 416-454.
[5] Joel Langer and Ron Perline, Local geometric invariants of integrable evolution equations, J. Math. Phys. 35 (1994), no. 4, 1732 – 1737. · Zbl 0801.58021
[6] Joel Langer and David A. Singer, The total squared curvature of closed curves, J. Differential Geom. 20 (1984), no. 1, 1 – 22. · Zbl 0554.53013
[7] Joel Langer and David A. Singer, Knotted elastic curves in \?³, J. London Math. Soc. (2) 30 (1984), no. 3, 512 – 520. · Zbl 0595.53001
[8] Richard S. Millman and George D. Parker, Elements of differential geometry, Prentice-Hall Inc., Englewood Cliffs, N. J., 1977. · Zbl 0425.53001
[9] 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. · Zbl 0531.53051
[10] U. Pinkall, Hopf tori in \?³, Invent. Math. 81 (1985), no. 2, 379 – 386. · Zbl 0585.53051
[11] Paul D. Scofield, Curves of constant precession, Amer. Math. Monthly 102 (1995), no. 6, 531 – 537. · Zbl 0881.53002
[12] Dirk J. Struik, Lectures on classical differential geometry, 2nd ed., Dover Publications, Inc., New York, 1988. · Zbl 0697.53002
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