ON ALMOST KAHLEM MANIFOLDS OF CONSTANT CURVATURE

An almost Hermitian manifold M = (M, J, g) is called an almost Kahler manifold if the corresponding Kahler form is closed (or equivalently <5jr,y,z0((Vjr/)r,Z) = O for X,Y,Ze%{M), where S and X(M) denotes the cyclic sum and the Lie algebra of all differentiable vector fields on M respectively).A Kahler manifold, which is defined by V/ = 0, is necessarily an almost Kahler manifold. It is well-known that an almost Kahler manifold with integrable almost complex structure is a Kahler manifold. A non-Kahler almost Kahler manifold is called a strictlyalmost Kahler manifold. Concerning the integrability of almost Kahler manifolds, the following conjecture by S. I. Goldberg is known ([2]):