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Some simple examples of almost Kähler non-Kähler structures. (English) Zbl 0858.53027

The main purpose of this paper is the construction of examples of strictly almost Kähler structures on some compact or non-compact manifolds. First, let \(M\) be an almost Kähler manifold and \(T = S^1 \times S^1\). Then the author proves that there exist uncountably many strictly almost Kähler structures on \(M \times T\), in particular with respect to the warped product \(M \times_f S^1 \times_{cf^{-1}} S^1\) where \(c\) is a constant and \(f\) is any non-constant, smooth, positive function on \(M\). Next, he uses his method to prove that on any Kähler manifold of dimension \(\geq 4\) there are uncountably many smooth non-equivalent strictly almost Kähler structures. Finally, he constructs uncountably many non-equivalent analytic strictly almost Kähler structures with negative constant scalar curvature on the torus \(T^6\). This construction may be extended to any \(T^{2k}\). Note that a strictly almost Kähler manifold is an almost Hermitian manifold \((M,g,J)\) with closed Kähler form but non-parallel structure \(J\).
The method used by the author leads to the construction of examples of compact almost Kähler manifolds which do not admit any Kähler structure.

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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References:

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