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On nearly-Kähler geometry. (English) Zbl 1020.53030

The paper deals with the canonical Hermitian connection of nearly-Kähler manifolds (shortly “NK manifolds”). Recall that an almost-Hermitian manifold \((M,g,J)\) is said to be nearly-Kähler if \((\nabla_{X}J)(X)=0\) for every \(X\in TM\) where \(\nabla\) is the Levi-Civita connection of \(g\) [A. Gray, cf. e.g. J. Differ. Geom. 4, 283-309 (1970; Zbl 0201.54401)]. An NK manifold is called strict if \(\nabla_{X}J\neq 0\) for every \(X\in TM, X\neq 0\).
The author starts with the observation that any complete, simply connected, NK manifold is the Riemannian product of a Kähler manifold and a strict NK manifold. Because of that, he restricts his considerations to strict NK manifolds. One of the main results proved in the paper states that if \((M,g,J)\) is a complete, strict NK manifold, then: (a) The canonical Hermitian connection has reduced holonomy provided the metric \(g\) is not Einstein; (b) The metric \(g\) has positive Ricci curvature (hence \(M\) is compact and with finite fundamental group); (c) The scalar curvature of \(g\) is a positive constant. The author proves also that if, in addition, \(M\) is simply connected and the holonomy group of the canonical Hermitian connection is contained in \(U(1)\times U(n-1)\), \(n=\frac{1}{2}\dim M\), then \(M\) is the Riemannian product of a strict NK manifold and the twistor space of a quaternionic-Kähler manifold with positive scalar curvature. This result has been earlier proved in the case when \(\dim M=6\) by F. Belgun and A. Moroianu [Ann. Global. Anal. Geom. 19, 307-318 (2001; Zbl 0992.53037)]. As a corollary, the author obtains a structure result for complete NK manifolds of dimension 10.

MSC:

53C29 Issues of holonomy in differential geometry
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C28 Twistor methods in differential geometry
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C55 Global differential geometry of Hermitian and Kählerian manifolds
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