Contact Geometry of Curves

TL;DR

This paper explores the contact geometry of curves in various geometries, providing explicit invariant data and algorithms for constructing differential invariants in Riemannian manifolds, with applications to specific spaces like hyperbolic space.

Contribution

It offers a comprehensive method to compute differential invariants of curves using contact geometry, extending Cartan's moving frames and providing explicit algorithms without integration.

Findings

Complete invariant data for curves in low-dimensional equi-affine geometry.

Explicit algorithms for constructing differential invariants in Riemannian manifolds.

Constructed invariants for curves in hyperbolic space and constant curvature metrics.

Abstract

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 Poincare 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.