Metric Lines in Engel-type Groups

TL;DR

This paper investigates the classification of metric lines in Engel-type Carnot groups within sub-Riemannian geometry, introducing a new sequence method to analyze these structures and providing initial classifications for groups of arbitrary rank.

Contribution

It introduces a novel sequence method for studying metric lines and classifies metric lines specifically in Engel-type Carnot groups of arbitrary rank.

Findings

Classified metric lines in Engel-type groups $ ext{Eng}(n)$

Developed a new sequence method for analyzing metric lines

Provided initial steps towards classifying metric lines in general Carnot groups

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.