3-structures with torsion (Q838018)

The main problem addressed to is to find an appropriate notion of 3-Sasakian connection with torsion, motivated by the theory of hyper-Kähler manifolds with torsion. To this end, one first establishes sufficient conditions for the integrability of the 3-dimensional distribution \({\mathcal V}\) spanned by the Reeb vector fields on an almost 3-contact manifold. Under the assumption that \({\mathcal V}\) is integrable and the manifold under consideration is endowed with a metric compatible with the almost contact structures, one introduces the concept of almost 3-contact metric connection with torsion. Several important properties of this geometric structure are established and the relationship to hyper-Kähler manifolds with torsion is pointed out.