Contact Geometry of Curves

Cartan's method of moving frames is briefly recalled in the context of immersed curves in the homogeneous space of a Lie group $G$. The contact geometry of curves in low dimensional equi-affine geometry is then made explicit. This delivers the complete set of invariant data which solves the $G$-equivalence problem via a straightforward procedure, and which is, in some sense a supplement to the equivariant method of Fels and Olver. Next, the contact geometry of curves in general Riemannian manifolds $(M,g)$ is described. For the special case in which the isometries of $(M,g)$ act transitively, it is shown that the contact geometry provides an explicit algorithmic construction of the differential invariants for curves in $M$. The inputs required for the construction consist only of the metric $g$ and a parametrisation of structure group $SO(n)$; the group action is not required and no integration is involved. To illustrate the algorithm we explicitly construct complete sets of differential invariants for curves in the Poincaré half-space $H^3$ and in a family of constant curvature 3-metrics. It is conjectured that similar results are possible in other Cartan geometries.