Leftinvariant PseudoRiemannian metrics on Lie groups: The null cone
Abstract
We study leftinvariant pseudoRiemannian 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 $\mu$ where zero is in the closure of the orbits: $0\in\overline{G\cdot \mu}$. 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 nontrivial Levidecomposable Lie algebras in the null cone are given.
 Publication:

arXiv eprints
 Pub Date:
 February 2024
 DOI:
 10.48550/arXiv.2402.03536
 arXiv:
 arXiv:2402.03536
 Bibcode:
 2024arXiv240203536H
 Keywords:

 Mathematics  Differential Geometry;
 Mathematical Physics;
 Mathematics  Group Theory
 EPrint:
 22pages, 1 figure