We present a necessary and sufficient condition for an Einstein metric on a symplectic manifold (M,ω) to be ω-compatible. An immediate consequence of the solution of the Goldberg conjecture by Sekigawa is that there exists no Einstein metric with non-negative Ricci curvature satisfying our condition on compact symplectic manifolds that admit no Kähler structure.
The first named author would like to thank the Department of Mathematics of the University of Botswana for its hospitality and support during the preparation of this paper. The authors also thank the referee for helping them improve the presentation.
© 2016 by De Gruyter