Spaces of harmonic maps of the projective plane to the four-dimensional sphere

TL;DR

This paper studies the structure of harmonic maps from the projective plane to the 4-sphere, revealing their existence conditions, connectivity, and explicit parameterizations for certain degrees using twistor lifts.

Contribution

It provides new results on the existence, connectivity, and explicit descriptions of harmonic maps from the projective plane to the 4-sphere, especially for degrees less than 6.

Findings

Spaces are empty for even harmonic degrees.

Spaces are path-connected for degrees less than 6.

Explicit parameterizations of canonical representatives are provided.

Abstract

The spaces of harmonic maps of the projective plane to the four-dimensional sphere are investigated in this paper by means of twistor lifts. It is shown that such spaces are empty in case of even harmonic degree. In case of harmonic degree less than 6 it was shown that such spaces are path-connected and an explicit parameterization of the canonical representatives was found. In addition, the last section summarizes known results for harmonic maps of the two-dimensional sphere to the four-dimensional sphere of harmonic degree less than 6.