On the Isometry Groups of Invariant Lorentzian Metrics on the Heisenberg Group

Abstract

This work concerns the invariant Lorentzian metrics on the Heisenberg Lie group of dimension three H3(R) and the bi-invariant metrics on the solvable Lie groups of dimension four. We start with the indecomposable Lie groups of dimension four admitting bi-invariant metrics and which act on H3(R) by isometries and we study some geometrical features on these spaces. On H3(R), we prove that the property of the metric being proper naturally reductive is equivalent to the property of the center being non-degenerate. These metrics are Lorentzian algebraic Ricci solitons