A sharp Eells-Sampson type theorem under positive sectional curvature upper bounds

We prove an extension of Eells and Sampson's rigidity theorem for harmonic maps from a closed manifold of non-negative Ricci curvature to a manifold of non-positive sectional curvature. We give an application of our result in the setting of harmonic-Einstein (or Ricci-harmonic) metrics and as a consequence we also recover a classical rigidity result of Hamilton for the problem of prescribed positive definite Ricci curvature.