Quaternion-Sasakian manifolds and reduction

An n-Sasakian manifold is a Riemannian manifold foliated by equidistant n-dimensional totally geodesic leaves such that the Riemann tensor is that of a curvature one space form on any triple of vector fields the include a fi eld everywhere tangent to the leaves of the foliation. Such manifolds are intimately connected to the parallel even Clifford orbifolds of Moroianu and Semmelmann.
We discuss an analogue of 3-Sasakian reduction in this setting. It turns out in this general setting actions amenable to reduction are somewhat sparse. However the Quaternion-Sasakian case (n=3) have a rich supply that include inhomo-geneous examples, reduction in this case closely ties back in with the 3-Sasakian reduction of Boyer, Galicki and Mann. We also touch on some examples of 7-Sasakian circle reduction.