  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 recover a classical rigidity result of Hamilton for the problem of
prescribed positive definite Ricci curvature.
