A splitting theorem for Kahler manifolds whose Ricci tensors have constanteigenvalues

It is proved that a compact Kahler manifold whose Ricci tensor has two dist inct constant non-negative eigenvalues is locally the product of two Kahler -Einstein manifolds. A stronger result is established for the case of Kahle r surfaces. Without the compactness assumption, irreducible Kahler manifold s with Ricci tensor having two distinct constant eigenvalues are shown to e xist in various situations: there are homogeneous examples of any complex d imension 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 d imension n greater than or equal to 3 with two negative eigenvalues; there are non-homogeneous examples of complex dimension 2 with one of the eigenva lues zero. The problem of existence of Kahler metrics whose Ricci tensor ha s two distinct constant eigenvalues is related to the celebrated (still ope n) conjecture of Goldberg [24]. Consequently, the irreducible homogeneous e xamples with negative eigenvalues give rise to complete Einstein strictly a lmost Kahler metrics of any even real dimension greater than 4.