## 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.

### MSC:

 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

### Keywords:

paraquaternionic structure; Levi-Cività connection