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Integrability of the twistor space for a hypercomplex manifold. (English) Zbl 0906.53048

A hyperkähler manifold is, by definition, a Riemannian manifold equipped with a smooth parallel action of the algebra of quaternions on its tangent bundle. There is a so-called twistor space associated to every hyperkähler manifold. Many of the differential-geometric properties of a hyperkähler manifold can be described in terms of holomorphic properties of its twistor space. An almost quaternionic manifold is a smooth manifold \(M\) equipped with a smooth action of the algebra \(H\) of quaternions on the tangent bundle \(\Theta (M)\) to \(M\).
Theorem 1: Let \(M\) be an almost quaternionic manifold and let \(X\) be its twistor space. The following conditions are equivalent. (1) For two algebra maps \(I,J: \mathbb{C}\rightarrow H\) such that \(I\neq J\) and \(\overline{I} \neq J\), the induced almost complex structures \(M_I,M_J\) on \(M\) are integrable. (2) For every algebra map \(I:\mathbb{C}\rightarrow H\), the induced almost complex structure \(M_I\) on \(M\) is integrable. (3) The almost complex structure on \(X\) is integrable.

MSC:

53C55 Global differential geometry of Hermitian and Kählerian manifolds
32L25 Twistor theory, double fibrations (complex-analytic aspects)
32Q15 Kähler manifolds
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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