Some examples of Dirac-harmonic maps
Résumé
We discuss a method to construct Dirac-harmonic maps developed by J. Jost, X. Mo and M. Zhu (in J. Jost, X. Mo, M. Zhu / Some explicit constructions of Dirac-harmonic maps, J. Geom. Phys. 2009, no. 11, 1512-1527). 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 J. Jost, X.~Mo, M. Zhu / Some explicit constructions of Dirac-harmonic maps, J. Geom. Phys. 2009, no. 11, 1512-1527), were fulfilled. We show that for isometric immersions into spaceforms, these conditions are fulfilled only under special assumptions.In several cases we show the existence of solutions.
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