Strictly abnormal geodesics with a degeneracy point in the interior of their domain

TL;DR

This paper investigates strictly abnormal geodesics in rank 2 sub-Riemannian manifolds, showing they can lose their length-minimizing property after metric changes, with implications for understanding optimality in these geometries.

Contribution

It introduces a method to prove length-minimizing properties of abnormal curves and demonstrates that strictly abnormal geodesics can become non-minimizing after metric modifications.

Findings

Strictly abnormal geodesics can cease to be locally length-minimizing after a metric change.

A new method is developed to establish length-minimizing properties of abnormal curves.

Abnormal curves with certain degeneracy points are studied in detail.

Abstract

In this article, we study abnormal curves in a family of sub-Riemannian manifolds of rank 2. We focus on abnormal curves whose lifts to the cotangent bundle annihilate, at an interior point of the domain, all Lie brackets of length up to three of vector fields tangent to the distribution. We present a method to prove that such curves are length-minimizing. Finally, we prove that strictly abnormal geodesics may cease to be locally length-minimizing after a change of the metric.