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TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
 

Summary: TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
S 0002-9947(00)02624-6
Article electronically published on June 13, 2000
SYMPLECTIC 4-MANIFOLDS
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.
1. Introduction
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 [16] 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
manifold.
On an oriented Riemannian 4-manifold, the Weyl tensor decomposes under the

  

Source: Apostolov, Vestislav - Département de mathématiques, Université du Québec à Montréal

 

Collections: Mathematics