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Generalizations of 3-Sasakian manifolds and skew torsion. (English) Zbl 1448.53052

An almost 3-contact manifold is a smooth \((4n+3)\)-dimensional manifold endowed with three almost contact structures \((\varphi_i, \xi_i, \eta_i)\), \(i=1,2,3\), satisfying the following conditions: \begin{align*} \varphi_k&=\varphi_i\varphi_j-\eta_j\otimes\xi_i=\varphi_j\varphi_i+\eta_i\otimes\xi_j; \\ \xi_k&=\varphi_i\xi_j=-\varphi_j\xi_i;\\ \eta_k&=\eta_i\circ\varphi_j=-\eta_j\circ\varphi_i \end{align*} for all even permutations \((i,j,k)\) of \((1,2,3)\).
The paper contains a review on almost contact and 3-contact metric manifolds as well as on the sphere of associated almost contact structures.
The most important and remarkable result of the work is the introduction of new classes of almost 3-contact metric manifolds.
It is proved that any 3-\((\alpha,\delta)\)-Sasakian manifold is hypernormal. This is a generalization of Kashiwada’s theorem stating that a 3-contact metric manifold is necessarily 3-Sasakian.
The so-called canonical connection of an almost 3-contact metric manifold is studied. Some important particular cases are selected.

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53B05 Linear and affine connections
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53D10 Contact manifolds (general theory)
53C27 Spin and Spin\({}^c\) geometry
32V05 CR structures, CR operators, and generalizations
22E25 Nilpotent and solvable Lie groups
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