On Einstein Lorentzian nilpotent Lie groups

[摘要] In this paper, we study Lorentzian left invariant Einstein metrics on nilpotent Lie groups. We show that if the center of such Lie groups is degenerate then they are Ricci-flat and their Lie algebras can be obtained by the double extension process from an abelian Euclidean Lie algebra. We show that all nilpotent Lie groups up to dimension 5 endowed with a Lorentzian Einstein left invariant metric have degenerate center and we use this fact to give a complete classification of these metrics. We show that if g is the Lie algebra of a nilpotent Lie group endowed with a Lorentzian left invariant Einstein metric with non zero scalar curvature then the center Z(g) of g is nondegenerate Euclidean, the derived ideal [g, g] is nondegenerate Lorentzian and Z(g) subset of [g, g]. We give the first examples of Ricci-flat Lorentzian nilpotent Lie algebra with nondegenerate center. (C) 2020 Elsevier B.V. All rights reserved.

[发布日期] 2020-12-01

[关键词] Einstein Lorentzian manifolds;Nilpotent Lie groups;Nilpotent Lie algebras