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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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