Non-existence of almost Kähler structure on hyperbolic spaces of dimension \(2n\) \((\geqq 4)\)
type: Item
instance of: scholarly article
title: Non-existence of almost Kähler structure on hyperbolic spaces of dimension \(2n\) \((\geqq 4)\)
zbMATH Open document ID: 0815.53054
DOI: 10.1007/BF01450489
author: Takashi Oguro
author: Kouei Sekigawa
published in: Mathematische Annalen
publication date: 1994-12-21T00:00:00Z
full work available at URL: https://eudml.org/doc/165255
review text: 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.
reviewed by: David E. Blair
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 704908
zbMATH Keywords: constant curvature
zbMATH Keywords: almost Kähler manifolds
zbMATH Keywords: non-Kähler
MaRDI profile type: Publication
cites work: Nonexistence of 4-Dimensional Almost Kaehler Manifolds of Constant Curvature
cites work: Integrability of Almost Kaehler Manifolds
cites work: Curvature identities for Hermitian and almost Hermitian manifolds
cites work: Q4159926
cites work: On some compact Einstein almost Kähler manifolds
OpenAlex ID: W1993624636
description: scientific article; zbMATH DE number 704908