The twisted G${}_2$ equation for strong G${}_2$-structures with torsion

Abstract

We discuss general properties of strong G${}_2$-structures with torsion and we investigate the twisted G${}_2$ equation, which represents the G${}_2$-analogue of the twisted Calabi-Yau equation for SU$(n)$-structures introduced by Garcia-Fernández - Rubio - Shahbazi - Tipler. In particular, we show that invariant strong G${}_2$-structures with torsion do not occur on compact non-flat solvmanifolds. This implies the non-existence of non-trivial solutions to the twisted Calabi-Yau equation on compact solvmanifolds of dimensions $4$ and $6$. More generally, we prove that a compact, connected homogeneous space admitting invariant strong G${}_2$-structures with torsion is diffeomorphic either to $S^3\times T^4$ or to $S^3\times S^3\times S^1$, up to a covering, and that in both cases solutions to the twisted G${}_2$ equation exist. Finally, we discuss the behavior of the homogeneous Laplacian coflow for strong G${}_2$-structures with torsion on these spaces.

Final version, to appear in Pure and Applied Mathematics Quarterly

Cited by in corpus (1)

• Some remarks on strong ${\mathrm{G}}_2$-structures with torsion