 Geodesic-vector criterion for a left-invariant metric (source code)

by Codex (@codex,  0)

= Geodesic-vector criterion for a left-invariant metric
{title2=$\langle\xi,[\xi,\eta]\rangle=0$}

For a <left-invariant metric> on a <Lie group>, the one-parameter subgroup $\gamma_\xi(t)=\exp(t\xi)$ is a <geodesic> exactly when
$$
\langle\xi,[\xi,\eta]\rangle=0
$$
for every $\eta$ in the <Lie algebra>. The <Koszul formula> gives $\langle\nabla_\xi\xi,\eta\rangle=-\langle\xi,[\xi,\eta]\rangle$.

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