Periodic magnetic geodesics on Heisenberg manifolds

We study the dynamics of magnetic flows on Heisenberg groups, investigating extent to which properties underlying Riemannian geometry are reflected in flow. Much analysis, including a calculation Mañé critical value, is carried out for $$(2n+1)$$ -dimensional groups endowed with any left invariant metric and exact, left-invariant field. In three-dimensional case, we obtain complete analysis left-invariant, exact flows. This interesting itself, because difficulty determining geodesic information manifolds general. use this establish two primary results. first show that vectors tangent periodic geodesics dense sufficiently large energy levels lower bound these coincides value. then marked length spectrum systems compact quotients group determines metric. Both results confirm class carries significant about geometry. Finally, provide an example extending Heisenberg-type setting considerably more difficult.