The fundamental theorem of curves and classifications in the Heisenberg groups

TL;DR

This paper establishes a fundamental theorem for horizontally regular curves in Heisenberg groups, introduces the concept of curve order, and classifies these curves based on p-curvatures and contact normality.

Contribution

It extends the fundamental theorem of curves to Heisenberg groups and completes the classification of horizontally regular curves in these groups.

Findings

Curve order characterizes the embedding in subgroups $H_k$

Two curves with same order are related by a rigid motion if they share p-curvatures and contact normality

Complete classification of horizontally regular curves in $H_n$ for all $n \\geq 1$

Abstract

We study the horizontally regular curves in the Heisenberg groups $H_n$. We show the fundamental theorem of curves in $H_n$ $(n\geq 2)$ and define the concept of the orders for horizontally regular curves. We also show that the curve $\gamma$ is of order $k$ if and only if $\gamma$ lies in $H_k$ but not in $H_{k-1}$ up to a Heisenberg rigid motion; moreover, two curves with the same order differ from a rigid motion if and only if they have the same p-curvatures and contact normality. Thus, combining with our previous work we have completed the classification of horizontally regular curves in $H_n$ for $n\geq 1$.