Einstein nilpotent Lie groups

Abstract

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 $s\ne 0$. Using the notion of nice Lie algebra, we give the first example of a left-invariant Einstein metric with $s\ne 0$ on a nilpotent Lie group. We show that nilpotent Lie groups of dimension $\le 6$ do not admit such a metric, and a similar result holds in dimension $7$ with the extra assumption that the Lie algebra is nice.

24 pages; v2, improved criterion for nonexistence added (Theorem 4.1), proofs simplified and contents reorganized accordingly, two references added; v3, corrected a constant in Theorem 3.8 and Corollary 3.10, minor correction in the proofs, added examples of Einstein metrics in any indefinite signature, presentation improved, three references added. To appear in J. Pure Appl. Algebra

References in corpus (2)

• Indefinite Einstein metrics on nice Lie groups
• The Ricci tensor of almost parahermitian manifolds

Cited by in corpus (8)

• Ricci-flat and Einstein pseudoriemannian nilmanifolds
• Construction of nice nilpotent Lie groups
• Indefinite Einstein metrics on nice Lie groups
• Pseudo-Kähler and pseudo-Sasaki structures on Einstein solvmanifolds
• The Ricci-flatness that lurks in weight
• Indefinite nilsolitons and Einstein solvmanifolds
• Left-invariant almost para-complex structures on six-dimensional nilpotent Lie groups
• A non-Standard Indefinite Einstein Solvmanifold