Yang, Bo; Zheng, Fangyang On curvature tensors of Hermitian manifolds. (English) Zbl 1408.53031 Commun. Anal. Geom. 26, No. 5, 1195-1222 (2018). Summary: In this article, we examine the behavior of the Riemannian and Hermitian curvature tensors of a Hermitian metric, when one of the curvature tensors obeys all the symmetry conditions of the curvature tensor of a Kähler metric. We will call such metrics \(G\)-Kähler-like or Kähler-like, for lack of better terminologies. Such metrics are always balanced when the manifold is compact, so in a way they are more special than balanced metrics, which drew a lot of attention in the study of non-Kähler Calabi-Yau manifolds. In particular we derive various formulas on the difference between the Riemannian and Hermitian curvature tensors in terms of the torsion of the Hermitian connection. We believe that these formulas could lead to further applications in the study of Hermitian geometry with curvature assumptions. Cited in 1 ReviewCited in 39 Documents MathOverflow Questions: chern connection vs levi-civita connection MSC: 53B35 Local differential geometry of Hermitian and Kählerian structures Keywords:balanced metrics; non-Kähler Calabi-Yau manifolds; Hermitian connection PDFBibTeX XMLCite \textit{B. Yang} and \textit{F. Zheng}, Commun. Anal. Geom. 26, No. 5, 1195--1222 (2018; Zbl 1408.53031) Full Text: DOI arXiv