The classification of ERP G 2 -structures on Lie groups
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The classification of ERP G2‑structures on Lie groups Jorge Lauret1 · Marina Nicolini1 Received: 17 October 2019 / Accepted: 21 March 2020 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract A complete classification of left-invariant closed G2-structures on Lie groups which are extremally Ricci pinched (i.e., d𝜏 = 61 |𝜏|2 𝜑 + 16 ∗ (𝜏 ∧ 𝜏) ), up to equivalence and scaling, is obtained. There are five of them, they are defined on five different completely solvable Lie groups and the G2-structure is exact in all cases except one, given by the only example in which the Lie group is unimodular. Keywords G2-structures · Extremally Ricci pinched · Laplacian solitons Mathematics Subject Classification 53C30 · 53C25 · 53C29
1 Introduction A G2-structure on a seven-dimensional differentiable manifold M is a differential 3-form 𝜑 on M which is positive (or definite), in the sense that 𝜑 (uniquely) determines a Riemannian metric g on M together with an orientation. Playing a role analogous in a way to that of almost Kähler structures in almost Hermitian geometry, closed G2-structures (i.e., d𝜑 = 0 ) have been studied by several authors (see, e.g., [6–8, 10, 17, 18]). They appear as natural candidates to be deformed via the Laplacian flow 𝜕t𝜕 𝜑(t) = Δ𝜑(t) toward a torsion-free G2 -structure (i.e., d𝜑 = 0 and d ∗ 𝜑 = 0 ) producing a Ricci-flat Riemannian metric with holonomy contained in G2 (see [16] for an account of recent advances). The torsion of a closed G2-structure 𝜑 is completely determined by the 2-form 𝜏 = − ∗ d ∗ 𝜑 , which in addition satisfies that d ∗ 𝜑 = 𝜏 ∧ 𝜑. The following remarkable curvature estimate for closed G2-structures on a compact manifold M was discovered by Bryant (see [6, Corollary 3]):
�M
scal2 ∗ 1 ≤ 3
�M
(1)
|Ric|2 ∗ 1,
* Jorge Lauret [email protected] Marina Nicolini [email protected] 1
FaMAF and CIEM, Universidad Nacional de Córdoba, 5000 Córdoba, Argentina
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J. Lauret, M. Nicolini
where scal and Ric , respectively, denote scalar curvature and Ricci tensor of (M, g). Bryant called extremally Ricci pinched (ERP for short) the structures at which equality holds in (1) (see [6, Remark 13]) and proved that they are characterized by the following neat equation:
d𝜏 = 16 |𝜏|2 𝜑 +
1 6
∗ (𝜏 ∧ 𝜏).
(2)
In the compact case, this is actually the only way in which d𝜏 can quadratically depend on 𝜏 (see [6, (4.66)]). A non-necessarily compact (M, 𝜑) satisfying (2) is also called ERP. The first examples of ERP G2-structures in the literature are homogeneous (i.e., the automorphism group Aut(M, 𝜑) ∶= {f ∈ Diff(M) ∶ f ∗ 𝜑 = 𝜑} acts transitively on M). In [6], Bryant gave the first example on the homogeneous space SL2 (ℂ) ⋉ ℂ2 ∕SU(2) , which indeed admits a compact locally homogeneous quotient (the only compact ERP structure known so far), a second one was found on a unimodular solvable Lie group in [14] and a curve was given in [11]. (Each structure in the curve is actually equivalent
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