Harmonic almost Hermitian structures

TL;DR

This paper surveys conditions under which compatible almost complex structures on Riemannian manifolds are harmonic sections or maps, focusing on classical structures like Atiyah-Hitchin-Singer and Eells-Salamon in four dimensions.

Contribution

It provides a comprehensive review of both old and new results on harmonic almost Hermitian structures, emphasizing specific classical examples.

Findings

Conditions for harmonicity of almost complex structures

Characterization of Atiyah-Hitchin-Singer and Eells-Salamon structures

Summary of recent advances in the field

Abstract

This is a survey of old and new results on the problem when a compatible almost complex structure on a Riemannian manifold is a harmonic section or a harmonic map from the manifold into its twistor space. In this context, a special attention is paid to the Atiyah-Hitchin-Singer and Eells-Salamon almost complex structures on the twistor space of an oriented Riemannain four-manifold.