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.


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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