Pluriharmonic maps into principal fiber bundle and vertical torsion (Q1202799)

Let \(\Sigma,P,M\) be Riemannian manifolds, and \(\pi:P\to M\) a Riemannian submersion with totally geodesic fibres. If a map \(\varphi:\Sigma\to P\) is harmonic, then it is vertically harmonic. Let \(\pi\) be a principal \(G\)-bundle with a connexion form \(\omega\). Assume that the metric on \(P\) has horizontal spaces whose metrics are the horizontal lifts of the metric on \(M\). Then \(\varphi\) is vertically harmonic iff the codifferential \(\delta(\varphi^*\omega)=0\). The author gives conditions in terms of harmonicity to insure that the vertical torsion vanishes. If \(\Sigma\) is a Hermitian manifold, he establishes analogous results characterizing vertical pluriharmonicity of a pluriharmonic map \(\varphi:\Sigma\to P\) in terms of the vertical torsion.