Non-existence of almost Kähler structure on hyperbolic spaces of dimension \(2n\) \((\geqq 4)\)
In [Proc. Am. Math. Soc. 21, 96-100 (1969; Zbl 0174.250)] \textit{S. I. Goldberg} conjectured that a compact almost Kähler-Einstein manifold is Kähler. The second author proved this conjecture under the additional assumption of non-negative scalar curvature [J. Math. Soc. Japan 39, 677- 684 (1987; Zbl 0637.53053)]. Since the conjecture is still open one can ask if an almost Kähler manifold of constant curvature must be Kähler, in which case it must also be flat. This was proved by \textit{Z. Olszak} [Bull. Acad. Polon. Sci., Sér. Sci. Math. Astron. Phys. 26, 139-141 (1978; Zbl 0379.53034)] in dimensions \(\geq 8\) and by the reviewer [Proc. Am. Math. Soc. 110, No. 4, 1033-1039 (1990; Zbl 0718.53028)] in dimension 4 leaving the question open in dimension 6. In the present paper the authors give a proof in all dimensions of the non-existence of non-Kähler almost Kähler manifolds of constant curvature. The problem is a local one and compactness does not play a role. The proof is by contradiction. If a non-Kähler almost Kähler manifold of constant curvature exists, it is easy to see that the curvature must be negative. Thus locally hyperbolic space would carry the structure. So consider the usual upper half space model of hyperbolic space with metric \(g\) of constant negative curvature and suppose there is an almost complex structure \(J\) such that \((J,g)\) is almost Kähler. A marvelous piece of difficult analysis follows leading to a contradiction.
scientific article; zbMATH DE number 704908