Communications in Analysis and Geometry
Volume 29 (2021)
Local description of Bochner-flat (pseudo-)Kähler metrics
Pages: 525 – 577
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.
The work of the first author was supported by the Russian Science Foundation (grant No. 17-11- 01303).
The second author thanks Deutsche Forschungsgemeinschaft (Research training group 1523: Quantum and Gravitational Fields), Friedrich-Schiller-Universität Jena and Leibniz Universität Hannover for partial financial support.
Received 25 September 2017
Accepted 24 September 2018
Published 10 May 2021