Lorentz Ricci Solitons on 3-dimensional Lie groups

The three-dimensional Heisenberg group H (3) has three left-invariant Lorentzian metrics g (1), g (2), and g (3) as in Rahmani (J. Geom. Phys. 9(3), 295-302 (1992)). They are not isometric to each other. In this paper, we characterize the left-invariant Lorentzian metric g (1) as a Lorentz Ricci Soliton. This Ricci Soliton g (1) is a shrinking non-gradient Ricci Soliton. We also prove that the group E(2) of rigid motions of Euclidean 2-space and the group E(1, 1) of rigid motions of Minkowski 2-space have Lorentz Ricci Solitons.