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

TL;DR

The paper investigates strictly abnormal geodesics in a family of rank-2 sub-Riemannian manifolds, focusing on interior degeneracy where the lift annihilates Lie brackets up to length three. It constructs a parametric model $(M,\mathcal{D},g)$ with $X_1 = \partial_{x_1}$, $X_2 = \partial_{x_2} + x_1^2 x_2^b \partial_{x_3}$, a curve $\gamma(t) = (0,t,0)$, and a metric depending on $a,b$ and $\alpha$, showing three main results: (i) for $b \le 2a$ with $b$ even, $\gamma|_{[-\varepsilon,\varepsilon]}$ remains a geodesic for any $\alpha$; (ii) a normality criterion asserts that $\gamma|_{[t_1,t_2]}$ is normal iff $\alpha(t)=0$ on that interval; (iii) if $\alpha \equiv 1$ and $b>4a+4$, $\gamma$ is not locally geodesic. The combination yields a strictly abnormal geodesic that is not locally minimizing under some metric, and the paper thus demonstrates metric dependence of abnormal geodesics and the possibility of losing local minimality after a metric change, contributing to the understanding of geodesic regularity in sub-Riemannian geometry.

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.

Figures (1)

• Figure 1: The curve $\pi\circ\omega:=\kappa*\gamma|_{[2\rho, \varepsilon]}*\sigma$.

Theorems & Definitions (14)

• Theorem 1.1
• Theorem 1.2
• Example 1.3
• Remark 2.1
• Theorem 2.2: Pontryagin Maximum Principle
• Remark 2.3
• Proposition 2.4
• proof
• Corollary 2.5
• proof : Proof of Theorem \ref{['thm:abnormal-normal']}