ODE trajectories as abnormal curves in Carnot groups

TL;DR

This paper demonstrates that polynomial ODE trajectories can be lifted to abnormal curves in Carnot groups, providing explicit constructions and new examples of such curves, and analyzing their concatenations.

Contribution

It introduces a method to construct abnormal curves in Carnot groups from polynomial ODE trajectories and shows that concatenations of abnormal curves also lift abnormally.

Findings

Existence of abnormal curves for polynomial ODE trajectories in Carnot groups

Explicit construction of covectors for abnormal curves

Concatenations of abnormal curves also have abnormal lifts

Abstract

We prove that for every polynomial ODE there exists a Carnot group where the trajectories of the ODE lift to abnormal curves. The proof defines an explicit construction to determine a covector for the resulting abnormal curves. Using this method we give new examples of abnormal curves in Carnot groups of high step. As a byproduct of the argument, we also prove that concatenations of abnormal curves have abnormal lifts.