Closed $G_2$-eigenforms and exact $G_2$-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 nG 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 six-dimensional nilpotent Lie algebras h admitting an exact SL.3; C/-structure or a half-flat SU.3/-structure .!; / with exact , respectively.

Exact G$$_{\mathbf{2}}$$-structures on compact quotients of Lie groups

“…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, ) , where G is a simply connected solvable Lie group with (2, 3)-trivial Lie algebra , namely b 2 ( ) = 0 = b 3 ( ) , Γ ⊂ G is a cocompact discrete subgroup (lattice), and 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.…”

“…Assume now that is an exact G 2 -structure on = ⋊ D ℝ , namely 𝜑 = d α for some α ∈ Λ 2 * . By [13], we know that if is strongly unimodular, then the solvable ideal is not nilpotent.…”

“…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, ) , where G is a simply connected solvable Lie group with (2, 3)-trivial Lie algebra , namely b 2 ( ) = 0 = b 3 ( ) , Γ ⊂ G is a cocompact discrete subgroup (lattice), and is an invariant exact G 2 -structure on Γ�G , namely it is induced by a left-invariant exact G 2 -structure on G .…”

“…some codimension-one unimodular ideal of , which must be solvable and non-nilpotent by [13], and some derivation D ∈ Der ( ) . The strongly unimodular condition on is then encoded into the derivation D, while the existence of an exact G 2 -structure on implies the existence of a certain type of SU(3)-structure on .…”