Author: Jaeman Kim 1
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  • 1 Department of Mathematics Education, Kangwon National University, Kangwon Do, 200-701 Korea
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Abstract  

The Riemannian version of the Goldberg-Sachs theorem says that a compact Einstein Hermitian surface is locally conformal Kähler. In contrast to the compact case, we show that there exists an Einstein Hermitian surface which is not locally conformal Kähler. On the other hand, it is known that on a compact Hermitian surface M4, the zero scalar curvature defect implies that M4 is Kähler. Contrary to the compact case, we show that there exists a non-Kähler Hermitian surface with zero scalar curvature defect.