ALMOST EINSTEIN-HERMITIAN MANIFOLDS

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In this paper, we show that every almost Einstein-Hermitian 4-manifold (i.e., almost Hermitian 4-manifold with -invariant Ricci tensor and harmonic Weyl tensor) is either Einstein or Hermitian. Consequently, we obtain that any almost Einstein-Hermitian 4-manifold which is not Einstein must be Hermitian and that every almost Einstein-Hermitian 4-manifold which is not Hermitian is Einstein. In contrast to the 4-dimensional case, there exists an almost Einstein-Hermitian manifold of dimension which is neither Einstein nor Hermitian.