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Metric-line classification conjecture for jet spaces

From papers

Let Jk(R,R)J^k(\mathbb{R},\mathbb{R}) be the space of kk-jets of real functions of one real variable, equipped with its sub-Riemannian distance. A geodesic γ:R→Jk(R,R)\gamma:\mathbb{R}\to J^k(\mathbb{R},\mathbb{R}) is a metric line when

∣a−b∣=dist⁡Jk(R,R)(γ(a),γ(b))|a-b|=\operatorname{dist}_{J^k(\mathbb{R},\mathbb{R})}(\gamma(a),\gamma(b))

for every compact interval [a,b]⊂R[a,b]\subset\mathbb{R}. The geodesics are classified by their reduced dynamics as line, xx-periodic, homoclinic, heteroclinic of the direct type, or heteroclinic of the turn-back type. Metric-line classification conjecture. The metric lines in Jk(R,R)J^k(\mathbb{R},\mathbb{R}) are precisely the geodesics of the following types: line, homoclinic, and heteroclinic of the direct type. This conjecture remains open for homoclinic geodesics; it concerns the classification of globally minimizing geodesics in jet spaces and excludes the xx-periodic and turn-back heteroclinic types.

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Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. The metric-line classification conjecture for jet spaces

    Let Jk(R,Rn)\mathcal{J}^k(\mathbb{R},\mathbb{R}^n) be the kk-jet space of curves, equipped with its sub-Riemannian structure. A metric line is a globally minimizing sub-Riemannian geodesic. The geodesics are classified as line, periodic, homoclinic, turn-back, or direct-type.

    Metric-line classification conjecture. The metric lines in Jk(R,Rn)\mathcal{J}^k(\mathbb{R},\mathbb{R}^n) are precisely the line, homoclinic, and direct-type geodesics.

    The paper proves two families of homoclinic metric lines in J2(R,R2)\mathcal{J}^2(\mathbb{R},\mathbb{R}^2) and notes that periodic and turn-back geodesics are not metric lines. The general classification, including the direct-type case, remains open.

    source: Daniella Catalá, Miriam Vollmayr-Lee and Alejandro Bravo-Doddoli, “Metric Lines in the Space of Curves”, arXiv:2511.21065 (2025).

Sources & referencesView supporting material

Primary source

Alejandro Bravo-Doddoli, “Metric lines in Jet Space”, arXiv:2205.06698 (2023).

carnot-groupsgeodesicsjet-spacessub-riemannian-geometry

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