Extremal curves of the total curvature in homogeneous 3-spaces

The space of curves which are extremal for the total curvature energy is well understood in isotropic homogeneous 3-spaces, said otherwise, spaces of constant curvature. In this paper we obtain that space of extremals in homogeneous 3-spaces whose isometry group has dimension four, that is, rotationally symmetric homogeneous 3-spaces. Most of the geometry in these spaces is governed by the existence of a unit Killing vector field, ξ, sometimes called the Reeb vector field, which turns the homogeneous 3-space into the source of a Riemannian submersion whose target space is a surface with constant curvature. Here, we show that a curve is an extremal of the total curvature energy if and only if ξ   lies into either the rectifying plane or the osculating plane along that curve. Then, we prove that every rotationally symmetric homogeneous 3-space, except H2×RH2×R, admits a real one-parameter class of extremals with horizontal normal (Lancret helices). The whole family of extremals is completed with a second class made up of those curves with horizontal binormal. In contrast with the first class, it appears in any rotationally symmetric space, with no exception, and it can be modulated in the space of real valued functions. We also work out geometric algorithms to solve the so called solving natural equations for extremals problem, allowing us to determine them explicitly in many cases. In addition, we solve the closed curve problem by showing the existence of two families of closed extremals. Namely, a rational one-parameter class of closed Lancret helices that appears at any rotationally symmetric homogeneous 3-space, except in H2×S1H2×S1, and a second class of extremals with horizontal binormal, which can be identified with the class of convex closed curves in the Euclidean plane. We also present a quantization principle, à la Dirac, for extremal values of the total curvature energy acting on closed curves in any rotationally symmetric homogeneous 3-space.