Harmonicity of the Atiyah-Hitchin-Singer and Eells-Salamon almost complex structures

TL;DR

This paper investigates specific four-dimensional Riemannian manifolds and identifies conditions under which certain almost complex structures induce harmonic maps from their twistor spaces.

Contribution

It characterizes when the Atiyah-Hitchin-Singer and Eells-Salamon structures produce harmonic maps on the twistor space of four-manifolds.

Findings

Identifies conditions for harmonicity of the structures

Provides classification of manifolds with harmonic twistor maps

Enhances understanding of twistor geometry in four dimensions

Abstract

In this paper we describe the oriented Riemannian four-manifolds $M$ for which the Atiyah-Hitchin-Singer or Eells-Salamon almost complex structure on the twistor space ${\mathcal Z}$ of $M$ determines a harmonic map from ${\mathcal Z}$ into its twistor space.