In this paper, we show that every almost Einstein-Hermitian 4-manifold (i.e., almost Hermitian 4-manifold with J-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.

almost Einstein-Hermitian 4-manifold, Einstein, Hermitian, almost Einstein-Hermitian manifold of dimensio

Received: October 22, 2024; Accepted: November 18, 2024; Published: December 24, 2024

How to cite this article: Jaeman Kim, Almost Einstein-Hermitian manifolds, JP Journal of Geometry and Topology 30(2) (2024), 119-130. https://doi.org/10.17654/0972415X24008

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

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