Closed G2-eigenforms and exact G2-structures

TY - JOUR

T1 - Closed G2-eigenforms and exact G2-structures

AU - Freibert, Marco

AU - Salamon, Simon

PY - 2021/12/3

Y1 - 2021/12/3

N2 - A study is made of left-invariant $\mathrm{G}_2$-structures with an exact 3-form on a Lie group $G$ whose Lie algebra $\mathfrak{g}$ admits a codimension-one nilpotent ideal $\mathfrak{h}$. It is shown that such a Lie group $G$ cannot admit a left-invariant closed $\mathrm{G}_2$-eigenform for the Laplacian and that any compact solvmanifold $\Gamma\backslash G$ arising from $G$ does not admit an (invariant) exact $\mathrm{G}_2$-structure. We also classify the seven-dimensional Lie algebras $\mathfrak{g}$ with codimension-one ideal equal to the complex Heisenberg Lie algebra which admit exact $\mathrm{G}_2$-structures with or without special torsion. To achieve these goals, we first determine the six-dimensional nilpotent Lie algebras $\mathfrak{h}$ admitting an exact $\mathrm{SL}(3,\mathbb{C})$-structure $\rho$ or a half-flat $\mathrm{SU}(3)$-structure $(\omega,\rho)$ with exact $\rho$, respectively.

AB - A study is made of left-invariant $\mathrm{G}_2$-structures with an exact 3-form on a Lie group $G$ whose Lie algebra $\mathfrak{g}$ admits a codimension-one nilpotent ideal $\mathfrak{h}$. It is shown that such a Lie group $G$ cannot admit a left-invariant closed $\mathrm{G}_2$-eigenform for the Laplacian and that any compact solvmanifold $\Gamma\backslash G$ arising from $G$ does not admit an (invariant) exact $\mathrm{G}_2$-structure. We also classify the seven-dimensional Lie algebras $\mathfrak{g}$ with codimension-one ideal equal to the complex Heisenberg Lie algebra which admit exact $\mathrm{G}_2$-structures with or without special torsion. To achieve these goals, we first determine the six-dimensional nilpotent Lie algebras $\mathfrak{h}$ admitting an exact $\mathrm{SL}(3,\mathbb{C})$-structure $\rho$ or a half-flat $\mathrm{SU}(3)$-structure $(\omega,\rho)$ with exact $\rho$, respectively.

KW - math.DG

KW - 53C10 (Primary) 53C30, 22E25 (Secondary)

UR - https://arxiv.org/abs/2101.10061

U2 - 10.4171/RMI/1315

DO - 10.4171/RMI/1315

M3 - Article

SN - 0213-2230

JO - REVISTA MATEMATICA IBEROAMERICANA

JF - REVISTA MATEMATICA IBEROAMERICANA

ER -