A Splitting Theorem for Kähler Manifolds with Constant Eigenvalues of the Ricci Tensor

It is proved that a compact Kähler manifold whose Ricci tensor has two distinct constant non-negative eigenvalues is locally the product of two Kähler-Einstein manifolds. A stronger result is established for the case of Kähler surfaces. Irreducible Kähler manifolds with two distinct constant eigenvalues of the Ricci tensor are shown to exist in various situations: there are homogeneous examples of any complex dimension n ≥ 2, if one eigenvalue is negative and the other is positive or zero, and of any complex dimension n ≥ 3, if the two eigenvalues are negative; there are non-homogeneous examples of complex dimension 2, if one of the eigenvalues is zero. The problem of existence of Kähler metrics whose Ricci tensor has two distinct constant eigenvalues is related to the celebrated (still open) Goldberg conjecture [24]. Consequently, the irreducible homogeneous examples with negative eigenvalues give rise to complete Einstein strictly almost Kähler metrics of any even real dimension greater than 4. 2000 Mathematics Subject Classification. Primary 53B20, 53C25