Cordero, Luis A.; Fernandez, M.; De Leon, Manuel Examples of compact non-Kähler almost Kähler manifolds. (English) Zbl 0575.53015 Proc. Am. Math. Soc. 95, 280-286 (1985). The main result of this paper is the construction of a family of compact almost Kähler manifolds \(M^{2(r+1)}\), \(r\geq 1\), which are not Kählerian. These manifolds are compact quotients of \(H(1,r)\times S^ 1\) by a discrete subgroup (H(1,r) being a generalized Heisenberg group). Each of the manifolds considered is an \((r+1)\)-torus bundle over an \((r+1)\)-torus and in this way, the examples are generalizations of Thurston’s example \((r=1)\). Using the first Betti number, it is shown that, for r odd, \(M^{2(r+1)}\) cannot admit any Kähler structure. For r even, the authors show that the given metric on the example is also not Kählerian. In a forthcoming paper, the first two authors and Gray will show that also for r even \(M^{2(r+1)}\) can have no Kähler structure. Reviewer: L.Vanhecke Cited in 3 ReviewsCited in 16 Documents MSC: 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 53C55 Global differential geometry of Hermitian and Kählerian manifolds 53C30 Differential geometry of homogeneous manifolds Keywords:locally affine manifolds; almost Kähler manifolds; Heisenberg group; Betti number PDFBibTeX XMLCite \textit{L. A. Cordero} et al., Proc. Am. Math. Soc. 95, 280--286 (1985; Zbl 0575.53015) Full Text: DOI