Canonical submersions in nearly Kähler geometry

Abstract 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- <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>α</m:mi> <m:mo>,</m:mo> <m:mi>δ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \left(\alpha ,\delta ) -Sasaki manifolds admit one-dimensional submersions onto nearly Kähler orbifolds. As a secondary application, we reprove that a given class of nearly Kähler manifolds submerges onto quaternionic Kähler manifolds. This new proof gives a direct expression for the quaternionic structure on the base.