Canonical Submersions in Nearly K\"ahler Geometry

We explore submersions introduced by reducible holonomy representations of connections with parallel skew torsion. A submersion theorem extending previous, less general, results is given. As our main application we show that parallel 3-$(\alpha,\delta)$-Sasaki manifolds admit 1-dimensional submersions onto nearly K\"ahler orbifolds. As a secondary application we reprove that a certain class of nearly K\"ahler manifolds submerges onto quaternionic K\"ahler manifolds. This new proof gives an direct expression for the quaternionic structure on the base.