Abnormal extremals of left-invariant sub-Finsler quasimetrics on four-dimensional Lie groups

TL;DR

This paper characterizes abnormal extremals in four-dimensional Lie groups equipped with left-invariant sub-Finsler quasimetrics, providing criteria for their strict abnormality based on Lie algebra structure and Minkowski functionals.

Contribution

It introduces a new criterion for strict abnormality of extremals in four-dimensional Lie groups with sub-Finsler quasimetrics, linking algebraic structure to geometric properties.

Findings

Explicit description of abnormal extremals in the setting

Criterion for strict abnormality based on structure constants and Minkowski functionals

Framework applicable to four-dimensional Lie groups with sub-Finsler structures

Abstract

Abnormal extremals on four-dimensional connected Lie groups with left-invariant sub-Finsler quasimetric, defined by a seminorm on a two-dimensional subspace of the Lie algebra generating the algebra, are found. In terms of structure constant of Lie algebra and supporting Minkowski function for the unit ball of seminorm on two-dimensional subspace of Lie algebra, defining a quasimetric, we establish a criterion for strict abnormality of these extremals.