Metric Lines in Engel-type Groups

TL;DR

The paper tackles the problem of identifying metric lines—globally minimizing sub-Riemannian geodesics—in Carnot groups, introducing the sequence method that reduces geodesic analysis to a magnetic space via a metabelian semidirect-product framework. It develops the magnetic-space reduction, cost maps, and a robust compactness-based sequence argument to certify when homoclinic geodesics yield metric lines. Applying this to the Engel-type group Eng$(n)$, the authors obtain a complete classification: metric lines in Eng$(n)$ are precisely line geodesics and $r$-homoclinic geodesics; small-oscillation and $r$-periodic geodesics are not metric lines, and abnormal lines that are metric lines must be line geodesics. The sequence method proves to be more versatile than prior optimal-synthesis or weak-KAM approaches, with potential to extend to arbitrary rank and dimension in metabelian Carnot groups and beyond, including jet-space settings and Euler-Elastica connections in planar projections.

Abstract

In the framework of sub-Riemannian Manifolds, a relevant question is: what are the \enquote{metric lines} (i.e., the isometric embedding of the real line)? This article presents a conjecture classifying the metric lines in Carnot groups and takes the first steps in answering this question for \enquote{arbitrary rank} Carnot groups. We classify the metric lines of the Engel-type groups $\Eng(n)$ (Theorem 1.2), whose sub-Riemannian structure is defined on a non-integrable distribution of rank $n+1$. Our approach is a new method, called the sequence method, which we began to develop to study metric lines in the jet space.

Figures (4)

• Figure 1.1: The images show the space $R^{3}$, with coordinate $(x_1,x_2,\theta_0)$, and the projections by $\pi$ of the homoclinic-geodesic $c(t)$.
• Figure 1.2: The images show two solutions to the Euler-Elastica problem in the $(x,\theta_0)$-plane. The first panel presents a generic solution, and the second displays the Euler-soliton solution.
• Figure 3.1: The images show the $(x,y)$-plane and the projections by $pr$ of the homoclinic-geodesic $c_h(t)$ and the sequence of minimizing geodesics $c_n(t)$.
• Figure 4.1: The first panel displays a typical $r$-periodic solution of an-harmonic oscillator. The last three panels show the Maxwell point in the plane $(x_1,x_2)$, $(t,y)$ and $(t,z)$, respectively.

Theorems & Definitions (96)

• Conjecture 1.1
• Theorem 1.2
• Definition 2.1
• Definition 2.2
• Lemma 2.3: Proposition 1, bravo2022geodesics
• Definition 2.4
• Lemma 2.5
• Proposition 2.6
• Definition 2.7
• Theorem 2.8: Background Theorem