## 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.

### MSC:

 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:

### References:

 [1] Goldberg, S.I.: Integrability of almost Kähler manifolds. Proc. Am. Math. Soc.21, 96-100 (1969) · Zbl 0174.25002 [2] Gray, A.: Curvature identities for Hermitian and almost Hermitian manifolds. Tôhoku Math. J.28, 601-612 (1976) · Zbl 0351.53040 [3] Gray, A.: The structure of nearly Kähler manifolds. Math. Ann.223, 233-248 (1976) · Zbl 0345.53019 [4] Hitchin, N.: Compact four-dimensional Einstein manifolds. J. Differ. Geom.9, 435-441 (1974) · Zbl 0281.53039 [5] Kobayashi, S., Nomizu, K.: Foundations of differential geometry. II. New York: Interscience 1969 · Zbl 0175.48504 [6] Kowalski, O.: Generalized symmetric spaces. Lect. Notes Math., Vol. 805. Berlin, Heidelberg, New York: Springer 1980 · Zbl 0431.53042 [7] Libermann, P.: Classification and conformal properties of almost Hermitian structures, János Bolyai 31. Differential Geometry, pp. 371-391. Budapest (Hungary), 1979 [8] Sawaki, S.: On almost-Hermitian manifolds satisfying a certain condition on the almost-complex structure tensor, Diff. Geom. in honor of Kentaro Yano, pp. 443-450. Tokyo: Kinokuniya 1972 [9] Sawaki, S.: Sufficient conditions for an almost-Hermitian manifold to be Kählerian. Hokkaido Math. J.1, 21-29 (1972) · Zbl 0253.53023 [10] Sekigawa, K.: Almost Hermitian manifolds satisfying some curvature conditions. Kodai Math. J.2, 384-405 (1979) · Zbl 0423.53030 [11] Singer, I.M., Thorpe, J.A.: The curvature of 4-dimensional Einstein spaces, Papers in honor of K. Kodaira, pp. 355-365. Princeton: Princeton University Press 1969 · Zbl 0199.25401 [12] Tachibana, S.: On automorphisms of certain compact almost-Hermitian spaces. Tôhoku Math. J.13, 179-185 (1961) · Zbl 0112.13803 [13] Thurston, W.P.: Some simple examples of symplectic manifolds. Proc. Am. Math. Soc.55, 467-468 (1976) · Zbl 0324.53031 [14] Tricerri, F., Vanhecke, L.: Curvature tensors on almost Hermitian manifolds. Trans. Am. Math. Soc.267, 365-398 (1981) · Zbl 0484.53014 [15] Watson, B.: New examples of strictly almost Kähler manifolds. Proc. Am. Math. Soc.88, 541-544 (1983) · Zbl 0517.53039 [16] Wu, W.T.: Sur la structure preque complexe d’une variété différentiable réelle de dimension 4. C.R. Acad. Sci. Paris227, 1076-1078 (1948) · Zbl 0037.10304 [17] Yano, K.: Differential geometry on complex and almost complex spaces. New York: Pergamon 1965 · Zbl 0127.12405
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.