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.


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