Metric Lines in the Space of Curves

Abstract

This paper investigates sub-Riemannian geodesics in 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 contributes to the broader classification of metric lines in jet spaces. Additionally, we present precise criteria that characterize when a sub-Riemannian geodesic in the $2$-jet space of plane curves is a metric line.