Sub-Riemannian LR geodesic flows on Heisenberg groups

Abstract

In this paper, we study the normal geodesic flow associated with a sub-Riemannian structure corresponding to a left-invariant Riemannian metric and a right-invariant distribution (LR structure) on the $(2n+1)$-dimensional Heisenberg group. We investigate the integrability properties of the corresponding LR system and establish a distinction between the low-dimensional and higher-dimensional cases. In dimensions greater than $5$, we prove that the LR system is integrable in the non-commutative sense, while in dimensions $3$ and $5$ we obtain the integrability in the commutative sense. We compare these results with the known integrability properties of the corresponding LL systems, associated with a left-invariant metric and a left-invariant distribution, highlighting similarities and differences between the two types of sub-Riemannian geodesic flows.