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Quaternionic manifolds. (English) Zbl 0534.53030

Metriche invarianti, applicazioni armoniche e questioni connesse, Conv. 1981, Symp. Math. 26, 139-151 (1982).
[For the entire collection see Zbl 0473.00008.]
By a particular choice of the structure group, the authors get an almost quaternion manifold modelled on the n-dimensional quaternion projective space \({\mathbb{H}}P^ n\). For such an almost quaternion manifold, it is shown that a simple characterization for the existence of a torsion-free connection on M which is almost quaternion is possible. Integrability along Newlander-Nirenberg theory for almost complex manifolds leads to, as is well known, a trivial situation since curvature of the almost quaternion manifold acts as an additional obstruction for integrability. The author gives one possible definition of integrability to avoid this triviality by overlooking the curvature obstruction of the Newlander- Nirenberg theory. In this theory of quaternion manifolds, he shows that the 2-sphere bundle, which coordinatizes the set of all almost complex structures resulting from the reduction of GL(4n,\({\mathbb{R}})\) to G as described at the beginning of the paper, is in fact a complex manifold. It is called the twistor space of M. Finally, he gives a few interesting examples and discusses briefly differential operators on his type of quaternion manifold.
Reviewer: M.Nagaraj

MSC:

53C10 \(G\)-structures
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C55 Global differential geometry of Hermitian and Kählerian manifolds

Citations:

Zbl 0473.00008