A note on flat metric connections with antisymmetric torsion

  • Agricola I
  • Friedrich T
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Abstract

In this short note we study flat metric connections with antisymmetric torsion T≠0. The result has been originally discovered by Cartan/Schouten in 1926 and we provide a new proof not depending on the classification of symmetric spaces. Any space of that type splits and the irreducible factors are compact simple Lie group or a special connection on S7. The latter case is interesting from the viewpoint of G2-structures and we discuss its type in the sense of the Fernández-Gray classification. Moreover, we investigate flat metric connections of vectorial type. © 2010 Elsevier B.V.

Author-supplied keywords

  • Flat connections
  • Skew-symmetric torsion

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Authors

  • Ilka Agricola

  • Thomas Friedrich

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