The Bochner tensor is the Kähler analogue of the conformal Weyl tensor. In this article, we derive local (i.e., in a neighbourhood of almost every point) normal forms for a (pseudo-)Kähler manifold with vanishing Bochner tensor. The description is pined down to a new class of symmetric spaces which we describe in terms of their curvature operators. We also give a local description of weakly Bochner-flat metrics defined by the property that the Bochner tensor has vanishing divergence. Our results are based on the local normal forms for c-projectively equivalent metrics. As a byproduct, we also describe all Kähler–Einstein metrics admitting a $c$-projectively equivalent one.

中文翻译：

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Bochner-flat（伪）Kähler指标的本地描述

Bochner张量是保形Weyl张量的Kähler类似物。在本文中，我们推导了带有假Bochner张量的（伪）Kähler流形的局部（即，几乎在每个点附近）正态形式。该描述被归结为一类新的对称空间，我们根据它们的曲率算符对其进行描述。我们还给出了由Bochner张量消失的特性定义的弱Bochner平坦度量的局部描述。我们的结果基于c投影等效度量的局部范式。作为副产品，我们还描述了所有接受Kähler-Einstein度量的指标，这些指标均接受$ c $的投影等价物。