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Einstein almost cokähler manifolds

We study an odd-dimensional analogue of the Goldberg conjecture for compact Einstein almost Kähler manifolds. We give an explicit non-compact example of an Einstein almost cokähler manifold that is not cokähler. We prove that compact Einstein almost cokähler manifolds with non-negative $*$-scalar curvature are cokähler (indeed, transversely Calabi-Yau); more generally, we give a lower and upper bound for the $*$-scalar curvature in the case that the structure is not cokähler. We prove similar bounds for almost Kähler Einstein manifolds that are not Kähler.