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 M 4, the zero scalar curvature defect implies that M 4 is Kähler. Contrary to the compact case, we show that there exists a non-Kähler Hermitian surface with zero scalar curvature defect.
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References
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Communicated by János Szenthe
This study is supported by Kangwon National University
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Kim, J. Some examples of Hermitian surfaces. Period Math Hung 64, 25–28 (2012). https://doi.org/10.1007/s10998-012-9025-4
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DOI: https://doi.org/10.1007/s10998-012-9025-4