On Einstein Hermitian manifolds II
Bull. Korean Math. Soc. 2009 Vol. 46, No. 2, 289-294
Printed March 1, 2009
Jaeman Kim
Kangwon National University
Abstract : We show that on a Hermitian surface $M$, if $M$ is weakly $*$-Einstein and has $J$-invariant Ricci tensor then $M$ is Einstein, and vice versa. As a consequence, we obtain that a compact $*$-Einstein Hermitian surface with $J$-invariant Ricci tensor is K\"{a}hler. In contrast with the $4$-dimensional case, we show that there exists a compact Einstein Hermitian $(4n+2)$-dimensional manifold which is not weakly $*$-Einstein.
Keywords : Hermitian surface, weakly $*$-Einstein, $J$-invariant Ricci tensor, Einstein, vice versa, $*$-Einstein, K\"{a}hler, compact Einstein Hermitian $(4n+2)$-dimensional manifold
MSC numbers : 53A30, 53B35, 53C25, 53C55, 53C56
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