Geodesic
Stability of Geodesic Vectors in Low-Dimensional Lie Algebras
Journal of Lie theory, Tome 32 (2022) no. 4, pp. 1111-1123
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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 $4$.
Classification : 53C30, 37D40, 34D20
Mots-clés : Geodesic vector, Lie algebra, Lyapunov stability 
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     author = {A. K. Nguyen and Y. 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/JLT_2022_32_4_JLT_2022_32_4_a9/}
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A. K. Nguyen; Y. 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/JLT_2022_32_4_JLT_2022_32_4_a9/