Lorentz geometry of 4-dimensional nilpotent Lie groups

Abstract

In the present work we classify, up to automorphism, left invariant Lorentz metrics on 4-dimensional nilpotent Lie groups 123R and $$G_4$$G4. We investigate their geometry, especially holonomy groups and decomposability of these metrics. Finally, we find projective classes of the metrics and prove all of these metrics are also left invariant.