  There are five unimodular simply connected three dimensional unimodular non
abelian Lie groups: the nilpotent Lie group $\mathrm{Nil}$, the special unitary
group $\mathrm{SU}(2)$, the universal covering group
$\widetilde{\mathrm{PSL}}(2,\mathbb{R})$ of the special linear group, the
solvable Lie group $\mathrm{Sol}$ and the universal covering group
$\widetilde{\mathrm{E}_0}(2)$ of the connected component of the Euclidean
group. For each $G$ among these Lie groups, we give explicitly the list of all
Lorentzian left invariant metrics on $G$, up to un automorphism of $G$.
Moreover, for any Lorentzian left invariant metric in this list we give its
Ricci curvature, scalar curvature, the signature of the Ricci curvature and we
exhibit some special features of these curvatures. Namely, we give all the
metrics with constant curvature, semi-symmetric non locally symmetric metrics
and the Ricci solitons.

Differential Geometry