Mathematische Annalen

, Volume 328, Issue 4, pp 711–748 | Cite as

On the holonomy of connections with skew-symmetric torsion

  • Ilka Agricola
  • Thomas FriedrichEmail author


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 2-parameter families of linear metric connections and spinorial connections defined by 4-forms with parallel spinors.


Manifold Riemannian Manifold Euclidian Space Integral Formula Parameter Family 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Institut für MathematikBerlinGermany

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