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Metric geometries over the split quaternions

A. S. Dancer·H. R. Jorgensen·A. F. Swann
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Abstract

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

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A. S. Dancer, H. R. Jorgensen, A. F. Swann. 2004-12-10. Metric geometries over the split quaternions. https://arxiv.org/abs/math/0412215

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