A SPLITTING THEOREM FOR KÄHLER MANIFOLDS WHOSE RICCI TENSORS HAVE CONSTANT EIGENVALUES

It is proved that a compact Kahler manifold whose Ricci tensor has twodistinct, constant, non-negative eigenvalues is locally the product of twoKahler-Einstein manifolds. A stronger result is established for the case ofKahler surfaces. Irreducible Kahler manifolds with two distinct, constanteigenvalues of the Ricci tensor are shown to exist in various situations: thereare homogeneous examples of any complex dimension n > 1, if one eigenvalue isnegative and the other positive or zero, and of any complex dimension n > 2, ifthe both eigenvalues are negative; there are non-homogeneous examples ofcomplex dimension 2, if one of the eigenvalues is zero. The problem ofexistence of Kahler metrics whose Ricci tensor has two distinct, constanteigenvalues is related to the celebrated (still open) Goldberg conjecture.Consequently, the irreducible homogeneous examples with negative eigenvaluesgive rise to complete, Einstein, strictly almost Kahler metrics of any evenreal dimension greater than 4.Comment: 18 pages; final version; accepted for publication in International Journal of Mathematic