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 $4$.
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Dept. of Mathematical and Physical Sciences, La Trobe University, Melbourne, Australia
An Ky Nguyen; Yuri Nikolayevsky. Stability of Geodesic Vectors in Low-Dimensional Lie Algebras. Journal of Lie Theory, Tome 32 (2022) no. 4, pp. 1111-1123. http://geodesic.mathdoc.fr/item/JOLT_2022_32_4_a9/
@article{JOLT_2022_32_4_a9,
author = {An Ky Nguyen and Yuri Nikolayevsky},
title = {Stability of {Geodesic} {Vectors} in {Low-Dimensional} {Lie} {Algebras}},
journal = {Journal of Lie Theory},
pages = {1111--1123},
year = {2022},
volume = {32},
number = {4},
url = {http://geodesic.mathdoc.fr/item/JOLT_2022_32_4_a9/}
}
TY - JOUR
AU - An Ky Nguyen
AU - Yuri Nikolayevsky
TI - Stability of Geodesic Vectors in Low-Dimensional Lie Algebras
JO - Journal of Lie Theory
PY - 2022
SP - 1111
EP - 1123
VL - 32
IS - 4
UR - http://geodesic.mathdoc.fr/item/JOLT_2022_32_4_a9/
ID - JOLT_2022_32_4_a9
ER -
%0 Journal Article
%A An Ky Nguyen
%A Yuri Nikolayevsky
%T Stability of Geodesic Vectors in Low-Dimensional Lie Algebras
%J Journal of Lie Theory
%D 2022
%P 1111-1123
%V 32
%N 4
%U http://geodesic.mathdoc.fr/item/JOLT_2022_32_4_a9/
%F JOLT_2022_32_4_a9
