## Nearly Kähler 6-manifolds with reduced holonomy.(English)Zbl 0992.53037

Nearly Kähler manifolds are non-integrable almost Kähler manifolds $$(M,J,g)$$ satisfying the equation $$(\nabla_X J)X=0$$, where $$\nabla$$ is the Levi-Civita connection of $$G$$. While the geometry of this connection, including its curvature properties, is rather well understood on a nearly Kähler manifold, the other objects which can be associated to the almost complex structure have been neglected.
The authors discuss the geometry of the Chern connection on a nearly Kähler manifold. They show that, up to homothety, the only complete, 6-dimensional, non-Kähler nearly Kähler manifold with canonical Hermitian connection having reduced holonomy are $$\mathbb{C}P^3$$ and the flag $$F(1,2)$$ with their nearly Kähler structure arising from the twistor construction (a result that was conjectured by Reyes-Carriøn in his PhD thesis, Oxford 1993). The proof is very elegant and combines techniques from foliations and twistor geometry.

### MSC:

 53C29 Issues of holonomy in differential geometry 53C28 Twistor methods in differential geometry 53C55 Global differential geometry of Hermitian and Kählerian manifolds
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