Left-invariant Einstein Metrics Research Articles

Indefinite Einstein metrics on nice Lie groups

Abstract We introduce a systematic method to produce left-invariant, non-Ricci-flat Einstein metrics of indefinite signature on nice nilpotent Lie groups. On a nice nilpotent Lie group, we give a simple algebraic characterization of non-Ricci-flat left-invariant Einstein metrics in both the class of metrics for which the nice basis is orthogonal and a more general class associated to order two permutations of the nice basis. We obtain classifications in dimension 8 and, under the assumption that the root matrix is surjective, dimension 9; moreover, we prove that Einstein nilpotent Lie groups of nonzero scalar curvature exist in every dimension ≥ 8 {\geq 8} .

On Einstein Lorentzian nilpotent Lie groups

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)⊂[g,g]. We give the first examples of Ricci-flat Lorentzian nilpotent Lie algebra with nondegenerate center.

Non-naturally reductive Einstein metrics on Sp(n)

We prove that Sp(2k + l) admits at least two non-naturally reductive Einstein metrics which are Ad(Sp(k) × Sp(k) × Sp(l))-invariant if k < l. It implies that every compact simple Lie group Sp(n) for n ⩾ 4 admits at least 2[(n–1)/3] non-naturally reductive left-invariant Einstein metrics.

On left-invariant Einstein metrics that are not geodesic orbit

In this article, we prove that compact simple Lie groups SO(n) (n>12) admit at least two left-invariant Einstein metrics that are not geodesic orbit, which gives a positive answer to a problem recently posed by Nikonorov.

Invariant Einstein metrics on certain compact semisimple Lie groups

In this paper, we study left invariant Einstein metrics on compact semisimple Lie groups. A new method to construct holonomy irreducible non-naturally reductive Einstein metrics on certain compact semisimple (non-simple) Lie groups is presented. In particular, we show that if G is a classical compact simple Lie group and H is a closed subgroup such that G/H is a standard homogeneous Einstein manifold, then there exist holonomy irreducible non-naturally reductive Einstein metrics on H×G, except for some very special cases. A further interesting result of this paper is that for any compact simple Lie group G, there always exist holonomy irreducible non-naturally reductive Einstein metrics on the compact semisimple Lie groups Gn, for any n≥4.

Left-invariant Einstein metrics on S3×S3

The classification of homogeneous compact Einstein manifolds in dimension six is an open problem. We consider the remaining open case, namely left-invariant Einstein metrics $g$ on $G = \mathrm{SU}(2) \times \mathrm{SU}(2) = S^3 \times S^3$. Einstein metrics are critical points of the total scalar curvature functional for fixed volume. The scalar curvature $S$ of a left-invariant metric $g$ is constant and can be expressed as a rational function in the parameters determining the metric. The critical points of $S$, subject to the volume constraint, are given by the zero locus of a system of polynomials in the parameters. In general, however, the determination of the zero locus is apparently out of reach. Instead, we consider the case where the isotropy group $K$ of $g$ in the group of motions is non-trivial. When $K\not\cong \mathbb{Z}_2$ we prove that the Einstein metrics on $G$ are given by (up to homothety) either the standard metric or the nearly K\"ahler metric, based on representation-theoretic arguments and computer algebra. For the remaining case $K\cong \mathbb{Z}_2$ we present partial results.

Compact simple Lie groups admitting left-invariant Einstein metrics that are not geodesic orbit

In this article, we prove that the compact simple Lie groups SU(n) for n≥6, SO(n) for n≥7, Sp(n) for n≥3, E6,E7,E8, and F4 admit left-invariant Einstein metrics that are not geodesic orbit. This gives a positive answer to an open problem recently posed by Nikonorov.

Non-naturally reductive Einstein metrics on $$\mathrm {SO}(n)$$ SO ( n )

In this article, we prove that every compact simple Lie group $${\mathrm S}{\mathrm O}(n)$$ for $$n\ge 10$$ admits at least $$2\left( [\frac{n-1}{3}]-2\right) $$ non-naturally reductive left-invariant Einstein metrics.

Non-naturally reductive Einstein metrics on exceptional Lie groups

Given an exceptional compact simple Lie group G we describe new left-invariant Einstein metrics which are not naturally reductive. In particular, we consider fibrations of G over flag manifolds with a certain kind of isotropy representation and we construct the Einstein equation with respect to the induced left-invariant metrics. Then we apply a technique based on Gröbner bases and classify the real solutions of the associated algebraic systems. For the Lie group G 2 we obtain the first known example of a left-invariant Einstein metric, which is not naturally reductive. Moreover, for the Lie groups E 7 and E 8 , we conclude that there exist non-isometric non-naturally reductive Einstein metrics, which are Ad ( K ) -invariant by different Lie subgroups K .

Curvature spectra of simple Lie groups

The Killing form \beta\ of a real (or complex) semisimple Lie group G is a left-invariant pseudo-Riemannian (or, respectively, holomorphic) Einstein metric. Let \Omega\ denote the multiple of its curvature operator, acting on symmetric 2-tensors, with the factor chosen so that \Omega\beta=2\beta. The result of Meyberg [8], describing the spectrum of \Omega\ in complex simple Lie groups G, easily implies that 1 is not an eigenvalue of \Omega\ in any real or complex simple Lie group G except those locally isomorphic to SU(p,q), or SL(n,R), or SL(n,C) or, for even n only, SL(n/2,H), where p\ge q\ge0 and p+q=n>2. Due to the last conclusion, on simple Lie groups G other the ones just listed, nonzero multiples of the Killing form \beta\ are isolated among left-invariant Einstein metrics. Meyberg's theorem also allows us to understand the kernel of \Lambda, which is another natural operator. This in turn leads to a proof of a known, yet unpublished, fact: namely, that a semisimple real or complex Lie algebra with no simple ideals of dimension 3 is essentially determined by its Cartan three-form.

Einstein metrics on compact Lie groups which are not naturally reductive

The study of left-invariant Einstein metrics on compact Lie groups which are naturally reductive was initiated by D’Atri and Ziller (Mem Am Math Soc 18, (215) 1979). In 1996 the second author obtained non-naturally reductive Einstein metrics on the Lie group SU(n) for n ≥ 6, by using a method of Riemannian submersions. In the present work we prove existence of non-naturally reductive Einstein metrics on the compact simple Lie groups SO(n) (n ≥ 11), Sp(n) (n ≥ 3), E 6, E 7, and E 8.