Abstract for Sfb Preprint No. 586

On the holonomy of connections with skew-symmetric torsion Ilka Agricola and Thomas Friedrich We investigate the holonomy group of a linear metric connection with skew-symmetric torsion. In case of the euclidian space and a constant torsion form, this group is always semisimple. It does not preserve any non-degenerated $2$-form or any spinor. Suitable integral formulas allow us to prove similar properties in case of a compact Riemannian manifold equipped with a metric connection of skew-symmetric torsion. On the Aloff-Wallach space $N(1,1)$, we construct families of connections admitting parallel spinors. Furthermore, we investigate the geometry of these connections as well as the geometry of the underlying Riemannian metric. Finally, we prove that any $7$-dimensional $3$-Sasakian manifold admits $\mathbb{P}^2$-parameter families of linear metric connections and spinorial connections defined by $4$-forms with parallel spinors.