Metric Lines in the Space of Curves

TL;DR

This paper studies sub-Riemannian geodesics in the jet space of plane curves, identifying two families of globally minimizing geodesics and establishing criteria for recognizing metric lines.

Contribution

It demonstrates the existence of two families of metric lines in the 2-jet space of plane curves and provides criteria to identify such geodesics.

Findings

Identified two distinct families of metric lines in the 2-jet space.

Established criteria for recognizing metric lines among geodesics.

Contributed to the classification of metric lines in jet spaces.

Abstract

This paper investigates sub-Riemannian geodesics within the jet space of curves. We establish the existence of two distinct families of metric lines, that is, globally minimizing geodesics, in the $2$-jet space of plane curves. This result provides an initial contribution toward the broader classification of metric lines in jet spaces. Additionally, we present precise criteria, which characterize when a sub-Riemannian geodesic in the $2$-jet space of plane curves can be identified as a metric line.