Harmonicity of the Atiyah-Hitchin-Singer and Eells-Salamon almost complex structures

In this paper we describe the oriented Riemannian four-manifolds $M$ for which the Atiyah-Hitchin-Singer or Eells-Salamon almost complex structure on the twistor space $\mathcal{Z}$ of $M$ determines a harmonic map from $\mathcal{Z}$ into its twistor space.