Some examples of Dirac-harmonic maps

We discuss a method to construct Dirac-harmonic maps developed by Jost et al. (J Geom Phys 59(11):1512-1527, 2009). The method uses harmonic spinors and twistor spinors and mainly applies to Dirac-harmonic maps of codimension 1 with target spaces of constant sectional curvature. Before the present article, it remained unclear when the conditions of the theorems in Jost et al. (2009) were fulfilled. We show that for isometric immersions into space forms, these conditions are fulfilled only under special assumptions. In several cases, we show the existence of solutions