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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

Citations:

Zbl 0174.25002
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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
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