On the completeness problem of homogeneous Lorentz 3-dimensional spaces, Part II

Abstract: We consider 3-dimensional solvable Lie groups given by a semi-direct product $\mathbb{R}\ltimes_A\mathbb{R}^2$, where $A$ is a nonzero real $2\times2$ diagonalizable matrix. The completeness problem has been addressed in the literature for two limiting cases: the unimodular Lie group $\mathrm{Sol}$ and the homothetic Lie group $\mathrm{Ho}$. In this work, we solve the completeness problem of left-invariant Lorentzian metrics on the remaining Lie groups of the given form. Moreover, we show that every 3-dimensional non-unimodular Lie group has an incomplete left-invariant metric.