Abnormal extremals of left-invariant sub-Finsler quasimetrics on four-dimensional Lie groups with three-dimensional generating distributions

TL;DR

This paper characterizes abnormal extremals in four-dimensional Lie groups with three-dimensional generating distributions under left-invariant sub-Finsler quasimetrics, providing criteria for strong abnormality based on algebraic structure.

Contribution

It identifies three-dimensional generating subspaces and establishes a criterion for strong abnormal extremals using Lie algebra structure constants and dual seminorms.

Findings

Classification of three-dimensional generating subspaces

Criterion for strong abnormality of extremals

Application to sub-Finsler quasimetrics on Lie groups

Abstract

We find three-dimensional subspaces of four-dimensional connected Lie algebras, generating these algebras, and abnormal extremals on connected Lie groups with these Lie algebras and with left-invariant sub-Finsler quasimetrics defined by seminorms on such subspaces. In terms of the structure constants of Lie algebras and dual seminorms, we establish a criterion for the strong abnormality of these extremals.