Some explicit constructions of Dirac-harmonic maps

We construct explicit examples of Dirac-harmonic maps (ϕ,ψ) between Riemannian manifolds (M,g)(M,g) and (N,g′)(N,g′) which are non-trivial in the sense that ϕϕ is not harmonic. When dimM=2, we also produce examples where ϕϕ is harmonic, but not conformal, and ψψ is non-trivial.