Globke and Nikolayevsky (2017) conjectured that every compact pseudo-Riemannian Einstein solvmanifold is a nilmanifold: if a compact manifold M=G/ΓM=G/\Gamma, the quotient of a connected simply-connected solvable Lie group GG by a lattice Γ\Gamma, carries an indefinite metric with Ricci tensor Ric=λg\mathrm{Ric}=\lambda g induced by an invariant Einstein scalar product on the Lie algebra of GG, then GG must in fact be nilpotent. They proved the statement for dim⁡M≤7\dim M\le 7 through a Gelfond–Schneider-type transcendence obstruction, showed it equivalent to extending that obstruction to matrices of all sizes, and observed that it would follow from Schanuel's Conjecture. The problem remains open.

Background

An Einstein manifold is a Riemannian or pseudo-Riemannian manifold (the latter allowing a metric of indefinite signature) whose Ricci tensor satisfies Ric=λg\mathrm{Ric}=\lambda g for a constant λ\lambda. A solvmanifold is a quotient G/ΓG/\Gamma of a connected simply-connected solvable Lie group GG by a lattice, i.e. a discrete subgroup Γ\Gamma with finite-volume quotient; a nilmanifold is the special case where GG is nilpotent. In the Riemannian-signature theory these notions are rigid: a compact Einstein solvmanifold with a positive-definite metric is necessarily a flat torus, while the rich noncompact Riemannian theory of Einstein solvmanifolds built on nilsolitons — nilpotent Lie groups with a left-invariant metric whose Ricci tensor satisfies Ric=λg+D\mathrm{Ric}=\lambda g+D for a constant λ\lambda and a derivation DD of the Lie algebra (Heber and subsequent work) — has no compact examples beyond that trivial case.

For indefinite metrics the situation changes qualitatively. A compact pseudo-Riemannian homogeneous space with a transitive solvable isometry group is, by a rigidity theorem of Baues and Globke, necessarily of the form G/ΓG/\Gamma with Γ\Gamma a lattice in GG, and the metric pulls back to a bi-invariant pseudo-Riemannian metric on GG, that is, to an ad-invariant symmetric bilinear form (a scalar product) ⟨⋅,⋅⟩\langle\cdot,\cdot\rangle on the Lie algebra g\mathfrak g, i.e. one satisfying ⟨[x,y],z⟩+⟨y,[x,z]⟩=0\langle[x,y],z\rangle+\langle y,[x,z]\rangle=0 for all x,y,z∈gx,y,z\in\mathfrak g. The analysis of Globke and Nikolayevsky shows that when this scalar product is Einstein and g\mathfrak g is solvable but not nilpotent, the Lie algebra admits a reduction controlled by a single real matrix XX of a specific block form, whose eigenvalues consist of real numbers λ1,…,λk\lambda_1,\dots,\lambda_k and non-real complex conjugate pairs ζ1,ζˉ1,…,ζm,ζˉm\zeta_1,\bar{\zeta}_1,\dots,\zeta_m,\bar{\zeta}_m satisfying the quadratic identity

λ12+⋯+λk2+2Re(ζ1)2+⋯+2Re(ζm)2−2Im(ζ1)2−⋯−2Im(ζm)2=0,\lambda_1^2+\cdots+\lambda_k^2+2\mathrm{Re}(\zeta_1)^2+\cdots+2\mathrm{Re}(\zeta_m)^2-2\mathrm{Im}(\zeta_1)^2-\cdots-2\mathrm{Im}(\zeta_m)^2=0,

and the existence of a lattice in GG forces some exponential exp⁡(tX)\exp(tX) to be conjugate to a matrix with rational entries (in SL(n,Q)\mathrm{SL}(n,\mathbb Q)). Their Lemma 5.3 excludes this for matrix size n≤5n\le 5 using the Gelfond–Schneider theorem (which states that for algebraic a≠0,1a\ne 0,1 and irrational algebraic bb, every value of aba^b is transcendental); this yields their Theorem 1.3: every compact pseudo-Riemannian Einstein solvmanifold of dimension ≤7\le 7 is a nilmanifold. The method does not extend to larger matrices, and the conjecture would follow in all dimensions from Schanuel's Conjecture (which asserts that for complex numbers α1,…,αd\alpha_1,\dots,\alpha_d linearly independent over Q\mathbb Q, the field Q(α1,…,αd,eα1,…,eαd)\mathbb Q(\alpha_1,\dots,\alpha_d,e^{\alpha_1},\dots,e^{\alpha_d}) has transcendence degree at least dd), an unproven statement in transcendental number theory. The authors prove that the conjecture is in fact equivalent to Lemma 5.3 holding for all nn, and note that a counterexample to that lemma for some n≥6n\ge 6 would explicitly produce a non-nilpotent solvable Lie group with an Einstein scalar product and a lattice.

The conjecture is not vacuous on the affirmative side: compact pseudo-Riemannian Einstein nilmanifolds do exist, for instance the 7-dimensional example of signature (3,4)(3,4) constructed by Fernández, Freibert and Sánchez, and Conti and Rossi produced Einstein metrics of nonzero scalar curvature on nilpotent Lie groups in every dimension ≥8\ge 8. Compact homogeneous Lorentzian Einstein solvmanifolds are flat (Globke–Nikolayevsky, Corollary 5.6), so the genuinely open range concerns indefinite signatures of index at least two.

Problem Statement

Prove or refute Conjecture 1.4 of Globke and Nikolayevsky: every compact pseudo-Riemannian Einstein solvmanifold is a nilmanifold.

Here a compact pseudo-Riemannian Einstein solvmanifold is a compact manifold MM with an indefinite metric gg satisfying Ric=λg\mathrm{Ric}=\lambda g for a constant λ\lambda, admitting a transitive isometric action of a connected solvable Lie group GG; by the Baues–Globke rigidity theorem such an MM is of the form M=G/ΓM=G/\Gamma with Γ\Gamma a lattice in the simply-connected group GG, and gg is induced by a bi-invariant pseudo-Riemannian Einstein metric on GG, equivalently by an ad-invariant Einstein scalar product on the Lie algebra g\mathfrak g. The conjecture asserts that g\mathfrak g is then necessarily nilpotent, so that MM is a nilmanifold.

References

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