Metric geometries over the split quaternions - Dancer, A; Jørgensen, H; Swann, A (2005)
We give an overview of some recent results in hypersymplectic and para-quaternionic Kähler geometry, and introduce the notion of split three-Sasakian manifold. In particular, we discuss the twistor spaces and Swann bundles of para-quaternionic Kähler manifolds. These are used to classify examples with a fully homogeneous action of a semi-simple Lie group, and to construct distinct para-quaternionic Kähler metrics from indefinite real analytic conformal manifolds. We also indicate how the theory of toric varieties gives rise to constructions of hypersymplectic manifolds.