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On the holonomy of connections with skew-symmetric torsion

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Abstract

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.

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Authors and Affiliations

  1. Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, Sitz: John-von-Neumann-Haus, Adlershof, D-10099, Berlin, Germany

    Ilka Agricola & Thomas Friedrich

Authors
  1. Ilka Agricola
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  2. Thomas Friedrich
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Correspondence to Thomas Friedrich.

Additional information

Mathematics Subject Classification (2000):53 C 25, 81 T 30

We thank Andrzej Trautman for drawing our attention to these papers by Cartan – see [27].

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Agricola, I., Friedrich, T. On the holonomy of connections with skew-symmetric torsion. Math. Ann. 328, 711–748 (2004). https://doi.org/10.1007/s00208-003-0507-9

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