On Riemannian four-manifolds and their twistor spaces: a moving frame approach

TL;DR

This paper investigates the twistor space of oriented Riemannian four-manifolds using a moving frame approach, revealing rigidity results and connections to nearly Kähler geometry in Einstein, non-self-dual settings.

Contribution

It demonstrates that first-order linear conditions on twistor space structures imply self-duality of the manifold and explores the nearly Kähler nature of certain Einstein twistor spaces.

Findings

First-order linear conditions enforce self-duality.

Most known rigidity results are recovered.

The twistor space of Einstein manifolds resembles nearly Kähler manifolds.

Abstract

In this paper we study the twistor space $Z$ of an oriented Riemannian four-manifold $M$ using the moving frame approach, focusing, in particular, on the Einstein, non-self-dual setting. We prove that any general first-order linear condition on the almost complex structures of $Z$ forces the underlying manifold $M$ to be self-dual, also recovering most of the known related rigidity results. Thus, we are naturally lead to consider first-order quadratic conditions, showing that the Atiyah-Hitchin-Singer almost Hermitian twistor space of an Einstein four-manifold bears a resemblance, in a suitable sense, to a nearly K\"ahler manifold.