Geometria Complessa e Geometria Differenziale
Geometria Complessa e Geometria Differenziale
home | mail | papers | authors | news | seminars | events | open positions | login

D. Conti - F. A. Rossi

Einstein nilpotent Lie groups

created by rossi on 11 Jan 2018
modified on 19 Nov 2018


Published Paper

Inserted: 11 jan 2018
Last Updated: 19 nov 2018

Journal: J. Pure Appl. Algebra
Volume: no. 3
Pages: 976-997
Year: 2019
Doi: 10.1016/j.jpaa.2018.05.010

ArXiv: 1707.04454 PDF


We study the Ricci tensor of left-invariant pseudoriemannian metrics on Lie groups. For an appropriate class of Lie groups that contains nilpotent Lie groups, we introduce a variety with a natural $\mathrm{GL}(n,\mathbb{R})$ action, whose orbits parametrize Lie groups with a left-invariant metric; we show that the Ricci operator can be identified with the moment map relative to a natural symplectic structure. From this description we deduce that the Ricci operator is the derivative of the scalar curvature $s$ under gauge transformations of the metric, and show that Lie algebra derivations with nonzero trace obstruct the existence of Einstein metrics with $s\neq0$. Using the notion of nice Lie algebra, we give the first example of a left-invariant Einstein metric with $s\neq0$ on a nilpotent Lie group. We show that nilpotent Lie groups of dimension $\leq 6$ do not admit such a metric, and a similar result holds in dimension $7$ with the extra assumption that the Lie algebra is nice.

Keywords: Ricci tensor, Moment map, Einstein pseudoriemannian metrics, Nilpotent Lie groups

Credits | Cookie policy | HTML 5 | CSS 2.1