Journal of Lie Theory

no. 4
Stability of Geodesic Vectors in Low-Dimensional Lie Algebras
Journal of Lie Theory, Volume 32 (2022) no. 4, pp. 1111-1123
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 $4$.
DOI: 10.5802/jolt.1267
Classification: 53C30, 37D40, 34D20
Keywords: Geodesic vector, Lie algebra, Lyapunov stability 
@article{JOLT_2022_32_4_a9,
     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},
     doi = {10.5802/jolt.1267},
     zbl = {1508.37047},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.1267/}
}
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A. K. Nguyen; Y. Nikolayevsky. Stability of Geodesic Vectors in Low-Dimensional Lie Algebras. Journal of Lie Theory, Volume 32 (2022) no. 4, pp. 1111-1123. doi: 10.5802/jolt.1267

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