Summary: TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Article electronically published on June 13, 2000
WITH HERMITIAN WEYL TENSOR
VESTISLAV APOSTOLOV AND JOHN ARMSTRONG
Abstract. It is proved that any compact almost K¨ahler, Einstein 4-manifold
whose fundamental form is a root of the Weyl tensor is necessarily K¨ahler.
An almost K¨ahler manifold is an almost Hermitian manifold for which the fun-
damental 2-form is closed, and therefore symplectic. If, additionally, the almost
complex structure is integrable, we have a K¨ahler manifold.
The initial motivation for this paper is the conjecture due to Goldberg  that
a compact almost K¨ahler, Einstein manifold is necessarily K¨ahler. This conjecture
is still far from resolved, however, a number of results have been proved, mainly
in dimension 4, when additional curvature conditions are added. Of course, these
additional curvature conditions are necessarily conditions on the Weyl tensor of the
On an oriented Riemannian 4-manifold, the Weyl tensor decomposes under the