Journal of Lie Theory 32 (2022), No. 4, 1111--1123
Copyright Heldermann Verlag 2022
Stability of Geodesic Vectors in Low-Dimensional 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
[Abstract-pdf]
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 $\dot{Y}=\ad^t_YY$ 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 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 $4$.
Keywords: Geodesic vector, Lie algebra, Lyapunov stability.
MSC: 53C30, 37D40, 34D20.
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