${\mathrm{G}}_2$-structures with parallel skew-symmetric torsion

Abstract

We classify $7$-dimensional Riemannian manifolds carrying a metric connection with parallel skew-symmetric torsion whose holonomy is contained in ${\mathrm{G}}_2$, up to naturally reductive homogeneous spaces and nearly parallel ${\mathrm{G}}_2$-structures. This extends and completes the classification initiated by Th. Friedrich in the cocalibrated case. Incidentally, we also obtain the list of $\mathrm{SU}(3)$ geometries with parallel skew-symmetric torsion, up to naturally reductive homogeneous spaces and nearly Kähler manifolds.

final version, 31 pages