Almost Kahler stuctures on the Riemannian product of a 3-dimensional hyperbolic space and a real line

Abstract

An almost Hermitian manifold $M=(M, J,g)$ is called an almost Kahler manifold if the Kahler form is closed (or equivalently $\mathfrak{C}_{x.y,z}g\nabla_{X}JY,$ $Z$ $=0$ for $X,$ $Y,$ $Z\in \mathfrak{X}(M)$ , where $\mathfrak{C}$ and $\mathfrak{X}(M)$ denotes the cyclic sum and the Lie algebra of all differentiable vector fields on $M$ respectively). A K\"ahler manifold, which is defined by $\nabla J=0$ , is necessarily an almost Kahler manifold. A non-K\"ahler almost K\"ahler manifold is called a strictly almost Kahler manifold. It is well- known that an almost K\"ahler manifold with integrable almost complex structure is a K\"ahler manifold. Conceming the integrability of almost Kahler manifolds, the following conjecture by S. I. Goldberg is known ([1]): C0NJECTURE. A compact.almost Kahler Einstein manifold is a Kahler manifold.