2002/06/10 by Mohamed Boucetta, Boucetta, Mohamed · 1 citation
Mathematics · Medicine · Physics and Astronomy · #53C30 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Ophthalmology and Eye Disorders #Symplectic Geometry (math.SG) #math.DG #math.SG #msc:53C30
paper · pdf · doi:10.48550/arxiv.math/0206102
13 pages
arxiv created 2002/06/10 · openalex publication_date 2002/06/10 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The notion of Poisson manifold with compatible pseudo-metric was introduced by the author in [1]. In this paper, we introduce a new class of Lie algebras which we call a pseudo-Rieamannian Lie algebras. The two notions are strongly related: we prove that a linear Poisson structure on the dual of a Lie algebra has a compatible pseudo-metric if and only if the Lie algebra is a pseudo-Riemannian Lie algebra, and that the Lie algebra obtained by linearizing at a point a Poisson manifold with compatible pseudo-metric is a pseudo-Riemannian Lie algebra. Furthermore, we give some properties of the symplectic leaves of such manifolds, and we prove that every Poisson manifold with compatible metric (every Riemann-Lie algebra) is unimodular. As a final, we classify all pseudo-Riemannian Lie algebras of dimension 2 and 3.