A construction of Einstein-Weyl spaces via LeBrun-Mason type twistor correspondence

TL;DR

This paper constructs numerous Einstein-Weyl structures on S^2 x R with closed space-like geodesics, using LeBrun-Mason twistor theory and perturbations of CP^1 x CP^1, expanding understanding of their geometry.

Contribution

It introduces a method to generate Einstein-Weyl spaces close to constant curvature models via twistor theory and holomorphic disk configurations.

Findings

Infinitely many Einstein-Weyl structures constructed

All space-like geodesics in these structures are closed

Structures are perturbations of the diagonal in CP^1 x CP^1

Abstract

We construct infinitely many Einstein-Weyl structures on $S^2 \times R$ of signature (-++) which is sufficiently close to the model case of constant curvature, and whose space-like geodesics are all closed. Such structures are obtained from small perturbations of the diagonal of $CP^1 \times CP^1$ using the method of LeBrun-Mason type twistor theory. The geometry of constructed Einstein-Weyl space is well understood from the configuration of holomorphic disks. We also review Einstein-Weyl structures and their properties in the former half of this article.