A splitting theorem for Kahler manifolds whose Ricci tensors have constant eigenvalues

It is proved that a compact Kahler manifold whose Ricci tensor has two distinct constant non-negative eigenvalues is locally the product of two Kahler-Einstein manifolds. A stronger result is established for the case of Kahler surfaces. Without the compactness assumption, irreducible Kahler manifolds with Ricci tensor having two distinct constant eigenvalues are shown to exist in various situations: there are homogeneous examples of any complex dimension n greater than or equal to 2 with one eigenvalue negative and the other one positive or zero; there axe homogeneous examples of any complex dimension n greater than or equal to 3 with two negative eigenvalues; there are non-homogeneous examples of complex dimension 2 with one of the eigenvalues zero. The problem of existence of Kahler metrics whose Ricci tensor has two distinct constant eigenvalues is related to the celebrated (still open) conjecture of Goldberg [24]. Consequently, the irreducible homogeneous examples with negative eigenvalues give rise to complete Einstein strictly almost Kahler metrics of any even real dimension greater than 4.