Sub-Riemannian and sub-Lorentzian geodesics of the oscillator groups

Abstract

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 ℝ³. 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 g and % k be the Lie algebras of G and K, respectively. We consider on g the usual nilpotent Lie group structure with center k and call it N( g,k). 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 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.

Closed geodesics on compact semi-Riemannian manifolds
Souheib Allout, Salah Chaib
arXiv16 Oct 2026PreprintarXiv licenceFull textGeometry & topologyarXiv
Sub-Riemannian geodesics in the affine-additive group
Elia Bubani, Ioannis D. Platis, Georgios I. Simantiras +1
arXiv2 Oct 2026PreprintCC BY 4.0Full textGeometry & topology Optimisation & controlarXiv
Topological Obstructions to Riemannian-Lorentzian Continuation
Nathalie E. Rieger, Ödül Tetik
arXiv25 Sep 2026PreprintCC BY 4.0Full textGeometry & topology Mathematical physicsarXiv
Algebraic and Symmetrized Affine star-Ricci Solitons for the Bott Connection on Three-Dimensional Lorentzian Lie Groups
Yuxin Ding
arXiv14 Oct 2026PreprintarXiv licenceFull textGeometry & topologyarXiv
On the structure of $2$-step nilpotent Lorentzian naturally reductive Lie groups
Brian Luporini, Silvio Reggiani, Francisco Vittone
arXiv30 Sep 2026PreprintarXiv licenceFull textGeometry & topologyarXiv

Cite

BibTeX RIS

BibTeX

@misc{molina2026subriemannian,
  title = {{Sub-Riemannian and sub-Lorentzian geodesics of the oscillator groups}},
  author = {Mauricio Godoy Molina and Marcos Salvai},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.12027},
  eprint = {2610.12027},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.12027},
}

RIS

TY  - UNPB
TI  - Sub-Riemannian and sub-Lorentzian geodesics of the oscillator groups
AU  - Mauricio Godoy Molina
AU  - Marcos Salvai
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.12027
UR  - https://arxiv.org/abs/2610.12027
AB  - 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 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 g and % k be the Lie algebras of G and K, respectively. We consider on g the usual nilpotent Lie group structure with center k and call it N( g,k). 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 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.
ER  -

Abstract

Related

Cite