The uniton numbers of the harmonic 2-spheres in \(Gr_2(\mathbb{C}^4)\) (Q1389856)

The paper starts with a brief review of the basic facts of the twistor theories of harmonic maps: (1) from \(S^2\) to \(U(n)\) and (2) from \(S^2\) to \(Gr_2 (\mathbb{C}^n)\) due to \textit{K. Uhlenbeck} [J. Differ. Geom., 30, No. 1, 1-50 (1989; Zbl 0677.58020)] and due to \textit{F. E. Burstall} and \textit{J. C. Wood} [J. Differ. Geom. 23, 255-297 (1986; Zbl 0588.58018)], respectively. The author investigates the case when the target is \(Gr_2(\mathbb{C}^4)\), based on the observation that the ``forward replacements'' of (2) represent a special case of the ``uniton transformations'' of (1). As a result, the paper provides a classification of all harmonic maps from \(S^2\) to \(Gr_2(\mathbb{C}^4)\) in terms of isotropy and uniton number.