Einstein-Weyl structures corresponding to diagonal K\"ahler Bianchi IX metrics

TL;DR

This paper systematically analyzes four-dimensional Einstein-Weyl spaces with diagonal Kähler Bianchi IX metrics, extending previous work by characterizing subclasses with constant conformal scalar curvature and exploring their geometric properties.

Contribution

It extends prior analyses by classifying Einstein-Weyl structures with specific curvature conditions and identifying their relation to conformally Einstein and extremal metrics.

Findings

Identifies Einstein-Weyl structures with constant conformal scalar curvature.

Provides explicit parameters for the subclass with conformally scalar flat metrics.

Shows the most general conformally Einstein metric is an extremal Kähler metric.

Abstract

We analyse in a systematic way the four dimensionnal Einstein-Weyl spaces equipped with a diagonal K\"ahler Bianchi IX metric. In particular, we show that the subclass of Einstein-Weyl structures with a constant conformal scalar curvature is the one with a conformally scalar flat - but not necessarily scalar flat - metric ; we exhibit its 3-parameter distance and Weyl one-form. This extends previous analysis of Pedersen, Swann and Madsen , limited to the scalar flat, antiself-dual case. We also check that, in agreement with a theorem of Derdzinski, the most general conformally Einstein metric in the family of biaxial K\"ahler Bianchi IX metrics is an extremal metric of Calabi, conformal to Carter's metric, thanks to Chave and Valent's results.