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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.

MSC:

53C40 Global submanifolds
53A04 Curves in Euclidean and related spaces
53A05 Surfaces in Euclidean and related spaces
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