A note on paraquaternionic manifolds. (English) Zbl 1152.53024

A paraquaternionic structure on a differentiable manifold \(M\) is a subbundle \(V\subset\text{End}(TM)\) of rank 3 such that for any point of \(M\), there is an open neighborhood \(U\) and a frame \(\{J_1,J_2,J_3\}\) of \(V\) on \(U\) such that for all \(\alpha\in \{1,2,3\}\), we have \(J^2_\alpha=-\varepsilon_\alpha\) and \(J_\alpha J_{\alpha+1}= -J_{\alpha+1} J_\alpha= \varepsilon_{\alpha+2} J_{\alpha+2}\), where \(\varepsilon_1= 1\), \(\varepsilon_2= \varepsilon_3= 1\). Moreover, \((M, g, V)\) is said to be a paraquaternionic Kähler manifold if \(V\) is parallel with respect to the Levi-Cività connection of a Riemannian metric \(g\) adapted to the paraquaternionic structure \(V\).
The paper under review investigates some properties of the paraquaternionic nearly Kählerian manifold. The main result provides that nearly – paraquaternionic Kähler manifold with \(\dim M\geq 8\), is a paraquaternionic Kähler manifold.


53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics