Communications in Analysis and Geometry

Volume 29 (2021)

Number 3

Local description of Bochner-flat (pseudo-)Kähler metrics

Pages: 525 – 577



Alexey V. Bolsinov (Department of Mathematical Sciences, Loughborough University, Loughborough, United Kingdom; and Faculty of Mechanics and Mathematics, Moscow State University and Moscow Center for Fundamental and Applied Mathematics, Moscow, Russia)

Stefan Rosemann (Institut für Differentialgeometrie, Leibniz Universität Hannover, Germany)


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