Closed $\mathrm{G}_2$-eigenforms and exact $\mathrm{G}_2$-structures

Abstract

A study is made of left-invariant G 2 -structures with an exact 3-form on a Lie group G whose Lie algebra g admits a codimension-one nilpotent ideal h. It is shown that such a Lie group G cannot admit a left-invariant closed G 2 -eigenform for the Laplacian and that any compact solvmanifold Γ\G arising from G does not admit an (invariant) exact G 2 -structure. We also classify the seven-dimensional Lie algebras g with codimension-one ideal equal to the complex Heisenberg Lie algebra which admit exact G 2 -structures with or without special torsion. To achieve these goals, we first determine the sixdimensional nilpotent Lie algebras h admitting an exact SL(3, C)-structure ρ or a half-flat SU(3)-structure (ω, ρ) with exact ρ, respectively.

Remarks on Exact G$_{2}$-Structures on Compact Manifolds

“…There are also several classes of exact G 2 -Structures on non-compact manifolds that have recently been produced. Examples on unimodular Lie algebras with vanishing third Betti number are considered by Fernandez, Fino, and Raffero [26], and examples which are closed G 2 -Eigenforms are considered by Friebert and Salamon [27]. It is worthwhile to note that there are no compact quotients of the unimodular Lie algebra examples, and it is known that a compact manifold cannot admit closed G 2 -eigenforms [12].…”

Exact G$_2$-structures on compact quotients of Lie groups

“…Assume now that ϕ is an exact G 2 -structure on g = s ⋊ D R, namely ϕ = dα for some α ∈ Λ 2 g * . By [13], we know that if g is strongly unimodular, then the solvable ideal s is not nilpotent. We can write α = α + β ∧ η, where α ∈ Λ 2 s * and β ∈ s * .…”

“…However, it is still not known whether exact G 2 -structures may occur on compact 7-manifolds. A negative answer to this problem was given in [10,13] in some special cases. In [10], M. Fernández and the first and third named author of this paper proved that there are no compact examples of the form (Γ\G, ϕ), where G is a simply connected solvable Lie group with (2, 3)-trivial Lie algebra g, namely b 2 (g) = 0 = b 3 (g), Γ ⊂ G is a cocompact discrete subgroup (lattice), and ϕ is an invariant exact G 2 -structure on Γ\G, namely it is induced by a left-invariant exact G 2 -structure on G. In [13], Freibert and Salamon showed that the same conclusion holds, more generally, when the Lie algebra of G admits a codimension-one nilpotent ideal.…”

“…. , e 6 ) be the basis of s * for which the structure equations areg −1 3,4 ⊕ g −1 3,4 = −e13 , e 23 , 0, −e 46 , e 56 , 0 . Let B = (e 1 , .…”

“…12 , 2e13 , 0, −e 45 , = −e 23 , −2e 12 , 2e 13 , 0, −e 45 , −µ e 46 , (1 + µ) e 47 , −1 < µ ≤ − = −e 23 , −2e 12 , 2e13 , −e 14 − e 25 − e 47 , e 15 − e 34 − e 57 , 2e 67 , 0 .…”