
Journal of Lie Theory 32 (2022), No. 4, 11111123 Copyright Heldermann Verlag 2022 Stability of Geodesic Vectors in LowDimensional Lie Algebras An Ky Nguyen Dept. of Mathematical and Physical Sciences, La Trobe University, Melbourne, Australia kyanduynguyen@gmail.com Yuri Nikolayevsky Dept. of Mathematical and Physical Sciences, La Trobe University, Melbourne, Australia y.nikolayevsky@latrobe.edu.au [Abstractpdf] A naturally parameterised curve in a Lie group with a left invariant metric is a geodesic, if its tangent vector lefttranslated to the identity satisfies the Euler equation $\dot{Y}=\ad^t_YY$ on the Lie algebra $\g$ 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 oneparameter 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 $4$. Keywords: Geodesic vector, Lie algebra, Lyapunov stability. MSC: 53C30, 37D40, 34D20. [ Fulltextpdf (149 KB)] for subscribers only. 