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Nearly Kähler geometry and Riemannian foliations. (English) Zbl 1041.53021

The author considers strict and complete nearly Kähler manifolds with the canonical Hermitian connection. The holonomy representation of the canonical Hermitian connection is studied. It is shown (theorem 1) that a strict and complete nearly Kähler manifold is locally a Riemannian product of homogeneous nearly Kähler spaces, twistor spaces over Kähler manifolds and 6-dimensional nearly Kähler manifolds. As an application the author obtains structure results for totally geodesic Riemannian foliations admitting a compatible Kähler structure (theorem 2). Finally a classification result for the homogeneous case, reducing a conjecture of Wolf and Gray to its 6-dimensional form, is obtained.

MSC:

53C12 Foliations (differential geometric aspects)
53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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