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 GL(n, 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 = 0.Using the notion of nice Lie algebra, we give the first example of a leftinvariant Einstein metric with s = 0 on a nilpotent Lie group. We show that nilpotent Lie groups of dimension ≤ 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.

Ricci-flat and Einstein pseudoriemannian nilmanifolds

“…By the above-mentioned result, we know that f (n) = 1 for n ≤ 5 and f (6) = 2 = f (7). In addition, in [6, Corollary 3.7], we proved that the number of inequivalent nice bases on a fixed nilpotent Lie algebra is at most countable.…”

“…Thus, the existence of a σ-diagonal metric with signature δ and Ric = 1 2 kId is equivalent to the existence of a vector X satisfying conditions (K), (H), (L σ ) and (7) for some σ-invariant v ∈ d n . For fixed X, Equation (7) has a solution in v if and only if the left-hand side is orthogonal to ker t M ∆ ; such a solution can always be assumed to be σ-invariant up to replacing v with 1 2 (v + σ(v)).…”

“…In this section we review our results concerning Einstein metrics of nonzero scalar curvature s (namely, solutions of (1) with λ = 0); the results stated here are contained in [6,4,7]. The existence of an Einstein metric with nonzero scalar curvature puts strong constraints on the Lie algebra, even outside of the nice context.…”

“…Examples of Ricci-flat nilpotent Lie algebras appear in the literature in particular contexts: four-dimensional ( [30]), bi-invariant ( [9,14,19]), nearly parakähler ( [5]), G * 2 -holonomy ( [13]), or 2-step ( [16]). The first example of an Einstein metric with nonzero scalar curvature on a nilpotent Lie algebra was constructed by the authors in [7].…”

“…Since [17], an effective approach used to study the Ricci operator on a nilpotent Lie algebra is to parametrize metric Lie algebras by fixing an orthonormal basis and letting the structure constants vary; the Ricci operator can then be interpreted as a moment map in the sense of geometric invariant theory ( [20]). In fact, the Ricci operator can also be viewed as a moment map in the sense of symplectic geometry (see [7]). This is true for every signature, although the convexity properties exploited in [20] appear not to hold in the indefinite case.…”