On a variational problem for curves in Lie sphere geometry

Let $\Lambda$ be the unit tangent bundle of the unit 3-sphere acted on transitively by the contact group of Lie sphere transformations. We study the Lie sphere geometry of generic curves in $\Lambda$ which are everywhere transversal to the contact distribution of $\Lambda$. By the method of moving frames, we prove that such curves can be parametrized by a Lie-invariant parameter, the Lie arclength, and that in this parametrization they are uniquely determined, up to Lie sphere transformation, by four local invariants, the Lie curvatures. We then consider the simplest Lie-invariant functional on generic transversal curves defined by integrating the differential of the Lie arclength. The corresponding Euler-Lagrange equations are computed and the critical curves are characterized in terms of their Lie curvatures. In our discussion, we adopt Griffiths' exterior differential systems approach to the calculus of variations.