Harmonic Mappings into non-negatively curved Riemannian manifolds

TL;DR

This paper develops a new theory for harmonic mappings into Riemannian manifolds with nonnegative sectional curvature, extending classical results and establishing conditions under which harmonic maps are totally geodesic.

Contribution

It proves that harmonic maps into nonnegatively curved manifolds are totally geodesic under Ricci curvature conditions, extending existing theories to complete manifolds.

Findings

Harmonic maps are totally geodesic if target has nonnegative sectional curvature and Ricci curvature condition holds.

Extension of results from compact to complete manifolds with nonnegative sectional curvature.

Derivation of new corollaries from the main theorems.

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.