Four-dimensional almost Kähler Einstein manifolds. (English) Zbl 0731.53026

Sekigawa, K.; Vanhecke, L.

Ann. Mat. Pura Appl., IV. Ser. 157, 149-160 (1990).

Let (M,J,g) be an almost Kähler manifold. The *-Ricci tensor is defined to be the trace of the map \(Z\to R(X,JZ)JY\), where R denotes the curvature tensor of M. M is said to be *-Einstein if the *-Ricci tensor is proportional to g. The authors prove that a 4-dimensional compact almost Kähler manifold which is Einstein and *-Einstein must be Kähler.

Reviewer: K.Ogiue (Tokyo)

MSC:

53B35	Local differential geometry of Hermitian and Kählerian structures
53C25	Special Riemannian manifolds (Einstein, Sasakian, etc.)

Keywords:

Einstein manifold; almost Kähler manifold; Ricci tensor

References:

[1]	Goldberg, S. I., Integrability of almost Kähler manifolds, Proc. Amer. Math. Soc., 21, 96-100 (1969) · Zbl 0174.25002
[2]	Gray, A.; Barros, M.; Naveira, A. M.; Vanhecke, L., The Chern numbers of holomorphic vector bundles and formally holomorphic connections of complex vector bundles over almost complex manifolds, J. Reine Angew. Math., 314, 84-98 (1980) · Zbl 0432.53050
[3]	Kobayashi, S.; Nomizu, K., Foundations of differential geometry, II (1969), New York: Interscience Publ., New York · Zbl 0175.48504
[4]	Kodaira, K., On the structure of complex analytic surfaces, I, Amer. J. Math., 86, 751-798 (1964) · Zbl 0137.17501
[5]	Libermann, P., Classification and conformal properties of almost Hermitian structures, Differential Geometry, János Bolyai Soc., 31, 371-391 (1979)
[6]	Olszak, Z., A note on almost Kähler manifolds, Bull. Acad. Pol. Soi. sér. Soi. Math. Astronom. Phys., 26, 139-141 (1978) · Zbl 0379.53034
[7]	Sekigawa, K., On some 4-dimensional compact Einstein almost Kähler manifolds, Math. Ann., 271, 333-337 (1985) · Zbl 0562.53032
[8]	Sekigawa, K., On some compact Einstein almost Kähler manifolds, J. Math. Soc. Japan, 39, 677-684 (1987) · Zbl 0637.53053
[9]	Sekigawa, K., On some 4-dimensional compact almost Hermitian manifolds, J. Ramanujan Math. Soc., 2, 101-116 (1987) · Zbl 0668.53019
[10]	Tachibana, S., On almost-analytic vectors in almost-Kählerian manifolds, Tôhoku Math. J., 11, 247-265 (1959) · Zbl 0090.38603
[11]	Thurston, W. P., Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc., 55, 467-468 (1976) · Zbl 0324.53031
[12]	Tricerri, F.; Vanhecke, L., Curvature tensors on almost Hermitian manifolds, Trans. Amer. Math. Soc., 267, 365-398 (1981) · Zbl 0484.53014
[13]	Yano, K., Differential geometry on complex and almost complex spaces (1965), New York: Pergamon Press, New York · Zbl 0127.12405