The differential geometry of curves in the Heisenberg groups

We study the horizontally regular curves in the Heisenberg groups H-n. We prove a fundamental theorem for curves in H-n (n >= 1) and define the order of horizontally regular curves. We also show that the curve gamma is of order k if and only if, up to a Heisenberg rigid motion, gamma lies in H-k 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 H-n for n >= 1. (C) 2017 Elsevier B.V. All rights reserved.