On some compact Einstein almost Kähler manifolds. (English) Zbl 0637.53053

In this paper the following theorem is proved: Let \(M=(M,J,<.,.>)\) be a compact Einstein almost Kähler manifold whose scalar curvature is non- negative. Then M is a Kähler manifold. This theorem gives a partial positive answer to the following well known conjecture: The almost complex structure of a compact Einstein almost Kähler manifold is integrable. [See S. I. Goldberg, Proc. Am. Math. Soc. 21, 96-100 (1969; Zbl 0174.250)]. In a previous paper [Math. Ann. 271, 333-337 (1985; Zbl 0562.53032)], the author had shown the same result but only in the case of \(\dim M=4.\)
Reviewer: L.A.Cordero


53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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