A remark on four-dimensional almost Kähler-Einstein manifolds with negative scalar curvature
Concerning the Goldberg conjecture, we will prove a result obtained by applying the result of Iton in terms of L2-norm of the scalar curvature.
On some compact Einstein almost Kähler manifolds
Soit M=(M 4 , J, ) une variete d'Einstein compacte presque de Kahler dont la courbure scalaire est non negative. Alors M est une variete de Kahler
Examples of compact non-Kähler almost Kähler manifolds
A parametrized family of compact non-Kihler almost Kihler manifolds M2r+2, r > 1, is constructed. For r odd, these manifolds are shown to have odd first Betti number, so they cannot be Kihlerian. Introduction. Few examples, compact or not, of non-Kaihler almost Kaihler manifolds are known. In fact, for the noncompact case only the tangent bundle (and some related tensor bundles) of nonflat Riemannian manifolds were known to admit such structure until B. Watson's report [10] of a large family of other noncompact examples; in 1976, W. P. Thurston [9] reported the existence of a compact manifold M4, defined as a 2-torus bundle over a 2-torus, which possesses an almost Kiihler structure but does not admit any Kiihlerian structure (an explicit description of that structure is given in [1]), and again B. Watson constructed more examples of this kind by considering products (M4 x S1) x (M4 x S1) x ... ; as far as we know, no more examples are known in the literature. In the present paper, a large family of compact almost Kaihler manifolds M2r+2, r > 1, which are non-Kahler, is constructed. Indeed, M2r+2 is firstly obtained as a compact complete locally affine manifold (Theorem 1.3), and this fact allows an easy computation of its first Betti number; secondly, M2r+2 is also described as an (r + 1)-torus bundle over an (r + 1)-torus, extending in this way Thurston's example to higher dimensions; and thirdly, we consider M2r+2 as a quotient of H(1, r) x S1 by a discrete subgroup, where H(1, r) is a generalized Heisenberg group, and then we construct the almost Kaihler structure on it as in Abbena's approach (which corresponds to the value r = 1); finally, we compute the curvature, the Ricci, the *-Ricci and the torsion tensors of the structure to check whether some identities, holding for a Kiihlerian structure, are verified here. Summing up, we have the following: (1) For r odd, M2r+2 has an almost Kaihler metric and, since its first Betti number is also odd (Corollary 1.4), it does not admit any Kaihlerian metric. (2) For r even, M2r+2 has an almost Kaihlerian metric, and this particular metric is not Kiihlerian. 1. Compact quotients of the generalized Heisenberg group. The generalized Heisenberg group H(1, r), r > 1, is the Lie group of real matrices of the Received by the editors October 22, 1984 and, in revised form, December 19, 1984. 1980 Mathemuaics Su*iect CMi Secondary 53C30, 22E25. Key u"ons and phrases. Almost Kihler manifolds, generalized Heisenberg groups, locally affine manifolds. (?)1985 American Mathematical Society 0002-9939/85 $1.00 + $.25 per page 280 This content downloaded from 207.46.13.83 on Fri, 21 Oct 2016 04:42:19 UTC All use subject to http://about.jstor.org/terms COMPACT NON-KAHLER ALMOST KAHLER MANIFOLDS 281 form Ir P T O 1 Q 0O 0 1 where P and T both are (r x 1)-column matrices, Q e R. H(1, r) is easily shown to be a connected and simply-connected two-step nilpotent Lie group of dimension 2r + 1. Indeed, H(1, r) is canonically diffeomorphic to R2r+1 through the map V): R2r+1 H(1, r) given by ir -tz X (1.1) ?k(X Y, Z) =O 1 Y 0O 0 1