Compatible Complex Structures on Twistor Spaces

TL;DR

This paper investigates complex structures on the twistor space of a Riemannian 4-manifold, linking metric properties of the base manifold to complex geometric properties of the twistor space.

Contribution

It characterizes compatible complex structures on twistor spaces and relates their properties to the metric features of the underlying 4-manifold.

Findings

Characterization of compatible complex structures on twistor spaces

Translation of metric properties of M into complex geometric properties of Z

Insights into the interplay between Riemannian and complex geometry

Abstract

Let (M,g) be a Riemannian 4-manifold. The twistor space Z->M is a CP1-bundle whose total space Z admits a natural metric h. The aim of this article is to study properties of complex structures on (Z,h) which are compatible with the CP1-fibration and the metric h. The results obtained enable us to translate some metric properties on M in terms of complex properties on its twistor space Z.