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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.

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.

Reviewer: Johann Davidov (Sofia)

### 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 |