Integrability of almost Kaehler manifolds.

*(English)*Zbl 0174.25002The tangent bundle of a nonflat Riemannian manifold is the only known example of a non Kählerian almost Kähler manifold, flatness being the integrability condition of the almost complex structure. These spaces are not compact. For those which are, it is a strong conjecture that if the almost hermitian metric is an Einstein metric the manifold is Kählerian. If \(M\) is Kählerian, its fundamental form has vanishing covariant derivative with respect to the Kähler metric \(g\), so the curvature transformation of \(g\) commutes with its almost complex structure \(J\).

The main result is the converse, viz., if the curvature transformation of the almost Kähler manifold \(M(J,g)\) commutes with \(J\), then \(M\) is a Kähler manifold.

The following corollaries follow easily:

(a) The curvature transformation of the almost Kähler structure of the tangent bundle \(T(M)\) commutes with the almost complex structure of \(T(M)\), if and only if, \(M\) is locally flat;

(b) An almost Kähler manifold of constant curvature is a Kähler manifold, if and only if, it is locally flat;

(c) A compact symmetric almost Kähler manifold is a Kähler manifold.

Moreover, if the Ricci 2-form of an almost Kähler manifold coincides with its “Chern 2-form”, then the manifold is Kählerian.

The method of proof is based on the observation that a harmonic form of constant length on a Riemannian manifold, which need not be compact, has zero covariant derivative, if and only if, the Bochner-Lichnerowicz quadratic form is non-negative.

The main result is the converse, viz., if the curvature transformation of the almost Kähler manifold \(M(J,g)\) commutes with \(J\), then \(M\) is a Kähler manifold.

The following corollaries follow easily:

(a) The curvature transformation of the almost Kähler structure of the tangent bundle \(T(M)\) commutes with the almost complex structure of \(T(M)\), if and only if, \(M\) is locally flat;

(b) An almost Kähler manifold of constant curvature is a Kähler manifold, if and only if, it is locally flat;

(c) A compact symmetric almost Kähler manifold is a Kähler manifold.

Moreover, if the Ricci 2-form of an almost Kähler manifold coincides with its “Chern 2-form”, then the manifold is Kählerian.

The method of proof is based on the observation that a harmonic form of constant length on a Riemannian manifold, which need not be compact, has zero covariant derivative, if and only if, the Bochner-Lichnerowicz quadratic form is non-negative.

Reviewer: S. I. Goldberg

##### MSC:

53C55 | Global differential geometry of Hermitian and Kählerian manifolds |

##### Keywords:

differential geometry
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\textit{S. I. Goldberg}, Proc. Am. Math. Soc. 21, 96--100 (1969; Zbl 0174.25002)

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##### References:

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[2] | Shigeo Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds. II, Tôhoku Math. J. (2) 14 (1962), 146 – 155. · Zbl 0109.40505 · doi:10.2748/tmj/1178244169 · doi.org |

[3] | Shun-ichi Tachibana and Masafumi Okumura, On the almost-complex structure of tangent bundles of Riemannian spaces, Tôhoku Math. J. (2) 14 (1962), 156 – 161. · Zbl 0114.38003 · doi:10.2748/tmj/1178244170 · doi.org |

[4] | K. Yano and S. Bochner, Curvature and Betti numbers, Annals of Mathematics Studies, No. 32, Princeton University Press, Princeton, N. J., 1953. · Zbl 0051.39402 |

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