On some 4-dimensional compact Einstein almost Kähler manifolds. (English) Zbl 0562.53032

An almost Kähler manifold is a manifold with a reduction of the structure group of the tangent bundle to \(U(n)\), and such that the corresponding Hermitian 2-form \(\Omega\) is closed. It is thus in particular a symplectic manifold. The author shows that a compact, 4-dimensional Einstein almost Kähler manifold with positive scalar curvature is necessarily a Kähler manifold. This answers, in this particular dimension, a conjecture raised by S. F. Goldberg [Proc. Am. Math. Soc. 21, 96–100 (1969; Zbl 0174.25002)]. The proof uses curvature identities for almost Kähler manifolds and inequalities for the Gauss-Bonnet integrand and Pontryagin form of a four-dimensional Einstein manifold to derive an inequality
\[ \int_{M}(\lambda +\frac{1}{4}\| \nabla \Omega \|^2)\| \nabla \Omega \|^2\le 0, \]
where \(4\lambda\) is the scalar curvature. Consequently the Hermitian form must be covariant constant if \(\lambda\) is positive.


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


Zbl 0174.25002
Full Text: DOI EuDML


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