Classification of left-invariant metrics on the Heisenberg group

In this study, we investigate the Riemannian and Lorentzian geometry of left-invariant metrics on the Heisenberg group H2n+1H2n+1, of dimension 2n+12n+1. We describe the space of all the left-invariant metrics of Riemannian and Lorentzian signatures up to automorphisms of the Heisenberg group. Thus, we classify quadratic forms of the corresponding signatures with respect to the action of the symplectic group. We also investigate the curvature properties and holonomy of these metrics. The most interesting is the Lorentzian metric with a parallel, null, central, left-invariant vector field. Rahmani proved that this metric is flat in the case of Heisenberg group H3H3. We show that this metric is not flat in higher dimensions.