Metric connections with parallel skew-symmetric torsion

A geometry with parallel skew-symmetric torsion is a Riemannian manifold carrying metric connection torsion. Besides the trivial case of Levi-Civita connection, geometries non-vanishing arise naturally in several geometric contexts, e.g. on reductive homogeneous spaces, nearly Kähler or G2-manifolds, Sasakian and 3-Sasakian manifolds, twistor spaces over quaternion-Kähler manifolds positive scalar curvature. In this paper we study local structure On every such one can define natural splitting tangent bundle which gives rise to submersion smaller dimension endowed some extra structure. We show how previously known examples fit into pattern, construct new examples. particular where above has principal bundle, give complete classification corresponding