Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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Bull. Korean Math. Soc. 2009; 46(2): 289-294

Printed March 1, 2009

https://doi.org/10.4134/BKMS.2009.46.2.289

Copyright © The Korean Mathematical Society.

On Einstein Hermitian manifolds II

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