Harmonic Maps into Grassmannians

Abstract

A harmonic map from a Riemannian manifold into a Grassmannian manifold is characterized by a vector bundle, a space of sections of the bundle and a Laplace operator [10]. This characterization can be considered a generalization of a theorem of Takahashi [11]. We apply our main result which generalizes a theorem of Do Carmo and Wallach [4] to describe moduli spaces of special classes of harmonic maps from compact reductive Riemannian homogeneous spaces into Grassmannians. As an application, we give an alternative proof of the theorem of Bando and Ohnita [1] which states the rigidity of the minimal immersion of the complex projective line into complex projective spaces. Moreover, a similar method yields rigidity of holomorphic isometric embeddings between complex projective spaces, which is part of Calabi’s result [2]. Finally, we give a description of moduli spaces of holomorphic isometric embeddings of the projective line into quadrics [9].