\(\textrm{G}_2\)-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 \(\textrm{G}_2\), up to naturally reductive homogeneous spaces and nearly parallel \(\textrm{G}_2\)-structures. This extends and completes the classification initiated by Th. Friedrich in the cocalibrated case. Incidentally, we also obtain the list of \(\textrm{SU}(3)\) geometries with parallel skew-symmetric torsion, up to naturally reductive homogeneous spaces and nearly Kähler manifolds.