Harmonic maps into homogeneous Riemannian manifolds: twistor approach

This paper deals with the twistor approach to the study of harmonic maps $\varphi\colon M\to N$ from Riemann surfaces $M$ to Riemannian manifolds $N$. Let $N$ be a given Riemannian manifold and let $Z$ be an almost complex manifold. The idea of the approach is to construct a so-called twistor bundle $\pi\colon Z\to N$ with the following property: the projection $\pi\circ\psi\colon M\to N$ of any almost holomorphic map $\psi\colon M\to Z$ is a harmonic map. For wide classes of Riemannian manifolds $N$ the twistor approach enables one to construct all harmonic maps $\varphi\colon M\to N$ in this way, thus reducing the original real problem of describing the harmonic maps into Riemannian manifolds to the complex problem of describing the almost holomorphic maps into almost complex manifolds. In this paper a detailed study is made of the following classes of homogeneous Riemannian manifolds $N$ to which the twistor approach can be applied: the compact Lie groups, the loop spaces of such groups, and the Grassmann manifolds, including the Hilbert Grassmannian.