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Metric geometries over the split quaternions. (English) Zbl 1178.53041

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.

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
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