Stability of geodesic vectors in low-dimensional Lie algebras

TL;DR

This paper classifies the stability of geodesic vectors in low-dimensional Lie algebras, providing a comprehensive understanding of their behavior in three- and four-dimensional cases with specific metric properties.

Contribution

It offers a complete classification of Lyapunov stable and unstable geodesic vectors in 3D and certain 4D metric Lie algebras, advancing the understanding of geodesic stability.

Findings

Classification of stable and unstable geodesic vectors in 3D Lie algebras

Extension of classification to unimodular 4D Lie algebras

Insight into the stability properties of geodesics in low-dimensional Lie groups

Abstract

A naturally parameterised curve in a Lie group with a left invariant metric is a geodesic, if its tangent vector left-translated to the identity satisfies the Euler equation $\dot{Y}=\operatorname{ad}^t_YY$ on the Lie algebra $\mathfrak{g}$ of $G$. Stationary points (equilibria) of the Euler equation are called geodesic vectors: the geodesic starting at the identity in the direction of a geodesic vector is a one-parameter subgroup of $G$. We give a complete classification of Lyapunov stable and unstable geodesic vectors for metric Lie algebras of dimension $3$ and for unimodular metric Lie algebras of dimension $4$.