Metric Lines in the Space of Curves

TL;DR

This work analyzes sub-Riemannian geodesics in the 2-jet space of plane curves, establishing the existence of two distinct families of metric lines in $\\mathcal{J}^2(\\mathbb{R},\\mathbb{R}^2)$. It develops a sequence-method framework, introducing a magnetic auxiliary space $\\mathbb{R}^5_{(a,b)}$ and a period map that encodes geodesic asymptotics, and proves metric-line status for the two candidate homoclinic families via Hadamard-type diffeomorphism arguments. A central result shows that, under a one-to-one period-map constraint, homoclinic geodesics yield globally minimizing lines; two main families are proven to be metric lines, while the full general three-parameter homoclinic family remains open. The approach blends Carnot-group symmetry, a reduced Hamiltonian analysis, and compactness arguments to reduce metric-line classification in jet spaces to a tractable asymptotic study with explicit polynomial pencils and period data. The findings advance the broader conjecture that metric lines correspond to asymptotically velocity-aligned geodesics, with potential implications for global optimality in sub-Riemannian jet-space settings.

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.

Figures (6)

• Figure 2.1: The panels illustrate the hill interval and the algebraic curves associated with momentum $\mu$.
• Figure 2.2: The panel displays the projection of the different types of geodesics in $\mathcal{J}^2(\mathbb{R},\mathbb{R}^2)$ to $\mathbb{R}^3$, with coordinates $(x,y_1,y_2)$
• Figure 3.1: The panels display the graphs of $\varrho_1(\vartheta_2)$ on the left and $\varrho_2(\vartheta_2)$ on the right, with $\vartheta_2$ varying over the interval $[-1,1]$.
• Figure 3.2: The panels display the graphs of $\varrho_3(\vartheta_2)$ on the left with $\vartheta_2$ ranging over $(-1,1)$, and the graph $\frac{\Delta(\vartheta_2)}{6a^2b^2}$ on the right with $\vartheta_2$ ranging over $(-1,0)$.
• Figure 3.3: The panels present numerical evidence supporting that $N_{(a,b)}$ is simply connected. Our results further indicate that $N_{(a,b)} = \mathbb{R}^2\setminus[0,\infty]\times \{0\}$ for $0<ab$.

Theorems & Definitions (58)

• Conjecture 1
• Theorem A
• Definition 2
• Definition 3
• Lemma 4: Proposition 1, bravo2022geodesics
• Conjecture 5
• Proposition 6: Proposition 2.6, bravododdoligeltype
• Definition 7
• Theorem 8: Theorem 1.2, BravoDoddoli2024
• Proposition 9