Left-invariant Pseudo-Riemannian metrics on Lie groups: The null cone

2024/02/05 by Sigbjørn Hervik, Hervik, Sigbjorn
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Group Theory (math.GR) #Mathematical Physics (math-ph)

paper · pdf · doi:10.48550/arxiv.2402.03536

openalex publication_date 2024/02/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study left-invariant pseudo-Riemannian metrics on Lie groups using the bracket flow of the corresponding Lie algebra. We focus on metrics where the Lie algebra is in the null cone of the G=O(p,q)-action; i.e., Lie algebras μ where zero is in the closure of the orbits: 0∈G⋅ μ. We provide examples of such Lie groups in various signatures and give some general results. For signatures (1,q) and (2,q) we classify all cases belonging to the null cone. More generally, we show that all nilpotent and completely solvable Lie algebras are in the null cone of some O(p,q) action. In addition, several examples of non-trivial Levi-decomposable Lie algebras in the null cone are given.

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