Sub-Riemannian and sub-Lorentzian geodesics of the oscillator groups
The oscillator groups are four-dimensional solvable Lie groups, extensions of the Heisenberg Lie group. We present them as circle bundles of the standard contact sub-Riemannian structure on $\mathbb{R}^{3}$. We define sub-Riemannian and sub-Lorentzian structures on them, describe their geodesics explicitly and determine which of them are periodic. We also study a generalization: Let $G$ be a semisimple Lie group and $K$ a compact subgroup such that $\left( G,K\right) $ is a symmetric pair of the compact or the noncompact type. Let $\mathfrak{g}$ and $% \mathfrak{k}$ be the Lie algebras of $G$ and $K$, respectively. We consider on $\mathfrak{g}$ the usual nilpotent Lie group structure with center $\mathfrak{k}$ and call it $N\left( \mathfrak{g},\mathfrak{k}\right)$. A suitable semidirect product with $\operatorname{Ad}\left( K\right) $ yields a solvable Lie group $\operatorname{Osc}\left( G,K\right) $, which generalizes the oscillator groups and is also quadratic (that is, it possesses bi-invariant metrics; in particular, a canonical one). We define nonholonomic pseudo-Riemannian structures on it and find their geodesics explicitly. In doing so, we obtain a result that may not be merely auxiliary: formulas for the monoparametric subgroups of $\operatorname{Osc}\left( G,K\right) $ and for the sub-Riemannian geodesics of $N\left( \mathfrak{g},\mathfrak{k}\right) $ with the standard left-invariant distribution $\mathcal{D}$. Moreover, when $G/K$ is compact with rank one, we realize $\operatorname{Osc}\left( G,K\right) $ as a Stiefel bundle of $\left( N\left( \mathfrak{g},\mathfrak{k}\right) ,\mathcal{D}\right) $, up to finite coverings.