Sesión Geometría, Topología y Teoría de LieSub-Riemannian and sub-Lorentzian geodesics of the oscillator groups
Marcos Salvai
FAMAF (Universidad Nacional de Córdoba) y CIEM (CONICET), Argentina - Esta dirección de correo electrónico está siendo protegida contra los robots de spam. Necesita tener JavaScript habilitado para poder verlo.
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 \(N\left( G,K\right)\). 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 be not merely auxiliary: formulas for the monoparametric subgroups of \(\operatorname{Osc}\left( G,K\right) \) and for the sub-Riemannian geodesics of \(N\left( G,K\right) \) 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 manifold of \(\left( N\left( G,K\right) ,\mathcal{D}\right) \), up to finite coverings.
Trabajo en conjunto con: Mauricio Godoy Molina (Departamento de Matemática y Estadística, Universidad de la Frontera, Temuco, Chile). The collaboration was funded by the IMU-Simons Research Program for Developing Countries..