A splitting theorem for Kähler manifolds with constant eigenvalues of the Ricci tensor

Vestislav Apostolov (1) , Tedi Draghici (2) , Andrei Moroianu (3)

1 DMS - Département de Mathématiques et de statistique [UdeM- Montréal]
2 FIU - Florida International University [Miami]
3 CMLS - Centre de Mathématiques Laurent Schwartz

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. Without the compactness assumption, irreducible Kähler 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 ≥ 2 with one eigenvalue negative and the other one positive or zero; there are homogeneous examples of any complex dimension n ≥ 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 Kähler metrics whose Ricci tensor has two distinct constant eigenvalues is related to the celebrated (still open) conjecture of Goldberg. 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.

Domaines: Géométrie différentielle [math.DG]

Vestislav Apostolov, Tedi Draghici, Andrei Moroianu. A splitting theorem for Kähler manifolds with constant eigenvalues of the Ricci tensor. International Journal of Mathematics, 2001, 12, pp.769-789. ⟨10.1142/S0129167X01001052⟩. ⟨hal-00126076⟩