On the role of abnormal minimizers in sub-Riemannian geometry

Abstract

Consider a sub-Riemannian geometry $(U,D,g)$ where $U$ is a neighborhood at 0 in ${\mathbb{R}}^n,$ $D$ is a rank-2 smooth $(C^{\infty }$ or $C^{\omega })$ distribution and $g$ is a smooth metric on $D$. The objective of this article is to explain the role of abnormal minimizers in SR-geometry. It is based on the analysis of the Martinet SR-geometry.