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.
Compact indefinite almost Kähler Einstein manifolds
In the present article, we extend the integral formula on a compact almost Kähler manifold with positive-definite metric to the one on a compact indefinite almost Kähler manifold and give its applications for some special indefinite almost Kähler Einstein manifolds taking the related problems to the indefinite analogy of Goldberg conjecture into consideration.
NEW EXAMPLES OF STRICTLY ALMOST KAHLER MANIFOLDS
A parametrized family of non-Kahler almost Kahler manifolds is con- structed as the product of solvable Lie groups with almost cosymplectic structures. A family of compact strictly almost Kahler manifolds whose cohomology is consistent with that of Kahler manifolds is similarly obtained. Almost Kahler manifolds are almost Hermitian manifolds which carry the sym- plectic structure given by their closed Kahler 2-form, $, but which are not neces- sarily complex (if complex, they are Kahler). Those whose almost complex structure 7 is not integrable are called strictly almost Kahler. Few examples of strictly almost Kahler manifolds are known. In fact, only the tangent and cotangent bundles (and some related tensor bundles) of nonflat Riemannian manifolds were known to possess such structures until recently. In 1976, W. Thurston (T) reported a strictly almost Kahler structure on a 2-torus bundle over a 2-torus. He verified its non-Kahler status by the oddness of its first Betti number. In (Go), S. I. Goldberg studied the conjecture that the almost complex structure on a compact Einstein almost Kahler manifold is integrable. While some progress has been made, particularly upon restricting the curvature tensor (SI, S2, S-K, Gr), the conjecture remains unresolved. Even in dimension four, where a normal form for the curvature of an Einstein manifold exists (Hi, J, S-T), it is open. One reason may be the paucity of examples of compact strictly almost Kahler manifolds. We announce here the construction of a large family of noncompact almost Kahler manifolds which are not Kahler and an interesting compact strictly almost Kahler 10-dimensional manifold which seeds the construction of (6« + 4)- dimensional spaces of like properties. We wish to thank Allen Broughton for illuminating discussions on solvable Lie groups. 1. Almost contact metric structures. Let (M2m+X, , |, tj, g) be an almost contact metric manifold (B). Define the fundamental 2-form $> in the usual way. Numerous distinct structures are definable on M in terms of $ and tj and their covariant derivatives. We are specifically interested in: (1) M is contact if $ = dr\; (2) M is normal if ( , ) + 2dri 8> £ = 0; (3) M is Sasakian if it is normal and contact; (4) M is almost cosymplectic if d$ — 0 and dri — 0; (5) M is cosymplectic if V = 0.