Dancer, A. S.; Jørgensen, H. R.; Swann, A. F. Metric geometries over the split quaternions. (English) Zbl 1178.53041 Rend. Semin. Mat., Torino 63, No. 2, 119-139 (2005). Summary: 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. Cited in 13 Documents MSC: 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 32L25 Twistor theory, double fibrations (complex-analytic aspects) 53A30 Conformal differential geometry (MSC2010) 53C30 Differential geometry of homogeneous manifolds 53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics 53D20 Momentum maps; symplectic reduction 53C55 Global differential geometry of Hermitian and Kählerian manifolds 57S15 Compact Lie groups of differentiable transformations PDFBibTeX XMLCite \textit{A. S. Dancer} et al., Rend. Semin. Mat., Torino 63, No. 2, 119--139 (2005; Zbl 1178.53041) Full Text: arXiv