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A geometry with parallel skew-symmetric torsion is a Riemannian manifold carrying a metric connection with parallel skew-symmetric torsion. Besides the trivial case of the Levi-Civita connection, geometries with non-vanishing parallel skew-symmetric torsion arise naturally in several geometric contexts, e.g. on naturally reductive homogeneous spaces, nearly Kähler or nearly parallel G2-manifolds, Sasakian and 3-Sasakian manifolds, or twistor spaces over quaternion-Kähler manifolds with positivedoi:10.14760/owp-2018-16 fatcat:ewrorllaq5ezjncxiiuhpbupyi