The completeness problem on 3-dimensional non-unimodular Lie groups

2025/04/15 by Chaib, Salah, Ana Cristina Ferreira, Ferreira, Ana Cristina
Mathematics · #53C50 #Differential Equations and Boundary Problems #Differential Geometry (math.DG) #FOS: Mathematics #Primary 53C22 #Secondary 53C30 #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2504.10998

openalex publication_date 2025/04/15 · openalex created_date 2025/10/14 · openalex updated_date 2026/07/28

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 ℝ \ltimes_A ℝ², where A is a real 2 × 2 matrix with nonzero trace. The case where A is not diagonalizable over 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 ℝ. Additionally, we show that, when A is diagonalizable over ℂ but not over 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.

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