Harmonic Mappings into non-negatively curved Riemannian manifolds

Abstract

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.