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arXiv · 2202.11894

Stability of geodesic vectors in low-dimensional Lie algebras

An Ky Nguyen·Yuri Nikolayevsky
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Abstract

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 $\dot{Y}=\operatorname{ad}^t_YY$ on the Lie algebra $\mathfrak{g}$ 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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An Ky Nguyen, Yuri Nikolayevsky. 2022-02-24. Stability of geodesic vectors in low-dimensional Lie algebras. https://arxiv.org/abs/2202.11894

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