The differential geometry of curves in the Heisenberg groups

We study the horizontally regular curves in the Heisenberg groups Hn. We prove a fundamental theorem for curves in Hn(n≥1) and define the order of horizontally regular curves. We also show that the curve γ is of order k if and only if, up to a Heisenberg rigid motion, γ lies in Hk but not in Hk−1; moreover, two curves with the same order differ in a rigid motion if and only if they have the same invariants: p-curvatures and contact normality. Thus, combining these results with our previous work [3] we get a complete classification of horizontally regular curves in Hn for n≥1.