Harmonicity of totally geodesic maps into nonpositively curved metric spaces (Q1882765)

A smooth map between two Riemannian manifolds is harmonic if the trace of its second fundamental form (called the tension field) vanishes. Thus a totally geodesic map, i.e. a map with second fundamental form identically zero, is clearly harmonic. If we consider a map \(\varphi:(M,g)\to(X,d_X)\) between a Riemannian manifold and a metric space, then the totally geodesic condition means that any geodesic in \(M\) is mapped to a geodesic in \(X\). In the paper under review the author considers different types of energies defined for maps between a Riemannian manifold and a metric space and proves that any totally geodesic map is harmonic with respect to the Korevaar-Schoen-type [\textit{N. J. Korevaar} and \textit{R. M. Schoen}, Commun. Anal. Geom. 1, No. 4, 561--659 (1993; Zbl 0862.58004)] and Cheeger-type [\textit{J. Cheeger}, Geom. Funct. Anal. 9, No. 3, 428--517 (1999; Zbl 0942.58018)] energies.