  Fifty years ago, Eells and Sampson have proved a famous theorem in which they
argued that any harmonic mapping $f:(M,g) \rightarrow (\bar{M},\bar{g})$ is
totally geodesic if $(M, g)$ is a compact manifold with the nonnegative Ricci
tensor and the section curvature of $(\bar{M},\bar{g})$ is nonpositive.
Moreover, other main results of the theory of harmonic mappings "in the large"
are the results on harmonic maps into nonpositively curved Riemannian
manifolds.
  In our paper we develop a theory of harmonic mappings into Riemannian
manifolds with nonnegative sectional curvature. In particular, we will prove
that any harmonic map between Riemannian manifolds $f:(M,g) \rightarrow
(\bar{M},\bar{g})$ is totally geodesic if the section curvature of
$(\bar{M},\bar{g})$ is nonnegative and $(M, g)$ is a compact manifold with the
Ricci tensor $Ric \geq f^{*}\bar{Ric}$ for the pullback $f^{*}\bar{Ric}$ of the
Ricci tensor $\bar{Ric}$ by $f$. The above scheme will be extended to a
harmonic mapping of a complete manifold to a manifold with the nonnegative
sectional curvature. Moreover, we will obtain interesting corollaries from our
results.
