## Integrability of almost Kaehler manifolds.(English)Zbl 0174.25002

The 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.
(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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### References:

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