Title：A classification theorem of horizontal curves in the Heisenberg groups
We study the horizontal curves in the Heisenberg groups and give a classification theorem. By introducing the concept for the order of horizontal curves, the curves can be distinguished from being ”spacial” or ”planar”. We conclude that the curves with the same order live in the same subspace; moreover, if the curves have the same p-curvature and contact normality, by the fundamental theorem of curves in the Heisenberg groups, then those are actually congruent to each other up to a Heisenberg rigid motion. An inductive process of dimensional reduction is also developed to classify the horizontal curves in the higher dimensional Heisenberg groups. This is a joint work with Hung-Lin Chiu and Xiuhong Feng.