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arXiv · 0911.1602

A note on flat metric connections with antisymmetric torsion

Ilka Agricola·Thomas Friedrich
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Abstract

In this short note we study flat metric connections with antisymmetric torsion $T \neq 0$. 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 $S^7$. The latter case is interesting from the viewpoint of $G_2$-structures and we discuss its type in the sense of the Fernandez-Gray classification. Moreover, we investigate flat metric connections of vectorial type.

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Ilka Agricola, Thomas Friedrich. 2009-11-09. A note on flat metric connections with antisymmetric torsion. https://arxiv.org/abs/0911.1602

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