Harmonic maps into nonnegatively curved manifolds and the soul theorem

Abstract

We study harmonic maps f: \((M,g)\rightarrow (\bar{M},\bar{g})\) satisfying the curvature inequality \(\textrm{Ric}_g \ge f^{*}\textrm{Ric}_{\bar{g}}\). Using the Bochner identity, we show that if the sectional curvature of the target manifold is nonnegative along the image of the map, then every harmonic map from a compact manifold is totally geodesic, and it is constant whenever the inequality is strict at some point. Combining this result with Perelman’s proof of the Soul Conjecture, we obtain a rigidity theorem for harmonic maps into complete noncompact manifolds with nonnegative sectional curvature. In particular, if the image of the map contains a point at which all sectional curvatures of the target are strictly positive, then the soul of the tangent is a point. We also analyze the equality case of the Bochner identity and show that the images of eigendirections corresponding to eigenvalues of \(f^{\bar{g}}\) span two-planes of zero sectional curvature in the tanget manifold.