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On hyper \(f\)-structures. (English) Zbl 0859.53019

Yano introduced the idea of \(f\)-structures to generalize the concept of almost complex and almost contact structures. In this paper we introduce almost quaternionic \(f\)-structures to generalize those of almost hypercomplex and almost contact 3-structures. In particular we define the hyper PS-manifolds, that fiber over hyper Kähler manifolds, and study some of their geometric properties. The quaternionic analog of the Heisenberg group is studied in detail for having this kind of structure. This example permits the construction of hypercomplex manifolds with strictly negative sectional curvature. We also partially study the other main example of hyper \(f\)-manifolds: the 3-Sasakians, and we prove that they embed in hyper Kähler manifolds as extrinsic spheres. We use this observation to construct generalized Hopf surfaces and Calabi-Eckmann manifolds.

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
30G35 Functions of hypercomplex variables and generalized variables
32G07 Deformations of special (e.g., CR) structures
32Q15 Kähler manifolds
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