The completeness problem on 3-dimensional non-unimodular Lie groups
Salah Chaib, Ana Cristina Ferreira
TL;DR
This work completes the geodesic-completeness classification of left-invariant Lorentzian metrics on 3-dimensional non-unimodular Lie groups, focusing on algebras $\mathfrak{h}(\lambda)$ arising from $A$ diagonalizable over $\mathbb{R}$ and normalized to $\operatorname{diag}(1,\lambda)$. Using the Euler–Arnold framework, metric-normal-form reductions via $\mathrm{Aut}(\mathfrak{h}(\lambda))$ yield explicit geodesic fields for a finite family of representatives $\mathcal{Q}_k$, whose dynamics are analyzed through invariant planes, energy first integrals, and idempotents. The authors prove a complete dichotomy for $0<|\lambda|<1$: a Lorentzian metric is geodesically complete precisely when $e_3$ is timelike and $e_2$ is not spacelike; moreover, if $e_3$ is timelike and $e_2$ is timelike, all trajectories are bounded, while $e_2$ lightlike permits unbounded nonstationary curves. In the limiting cases $\lambda=\pm1$, they recover known results (complete for certain $\mathfrak{h}(-1)$ forms, incomplete for others) and finally show that every 3D non-unimodular Lie algebra possesses an incomplete left-invariant Lorentzian metric, extending the incompleteness phenomenon beyond the diagonalizable-over-$\mathbb{R}$ setting.
Abstract
We consider the completeness problem for left-invariant Lorentzian metrics on 3-dimensional non-unimodular Lie groups, all of which have Lie algebra of the form $\mathbb{R} \ltimes_A \mathbb{R}^2$, where $A$ is a real $2 \times 2$ matrix with nonzero trace. The case where $A$ is not diagonalizable over $\mathbb{C}$ was addressed in previous work by the authors, and the limiting case where $A$ is a scalar multiple of the identity is also known from the literature. In this paper, we determine all geodesically (in)complete left-invariant Lorentzian metrics for all other cases where $A$ is diagonalizable over $\mathbb{R}$. Additionally, we show that, when $A$ is diagonalizable over $\mathbb{C}$ but not over $\mathbb{R}$, there exists at least one incomplete metric. As a consequence of prior work and our results, we obtain that every 3-dimensional non-unimodular Lie group admits an incomplete left-invariant Lorentzian metric.