Metric connections with parallel torsion
The torsion of every metric connection on a Riemannian manifold has three components: one totally skew-symmetric, one of vectorial type, and one of twistorial type. In the first part of the talk I will explain the classification of complete simply connected Riemannian manifolds carrying a metric connection whose torsion is parallel, has non-zero vectorial component and vanishing twistorial component. In the second part I will describe the case where the only non-vanishing component of the torsion is totally skew-symmetric. Although apparently simpler than the previous case, the situation here is much more involved and a complete classification is currently not available.
The talk is based on joint works with Mihaela Pilca and Uwe Semmelmann.
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