Some sufficient conditions for a map to be harmonic (Q2714311)

\textit{J. Eells} and \textit{L. Lemaire} [Two reports on harmonic maps, World Scientific, Singapore (1995; Zbl 0836.58012)] claimed that a map from a compact Riemannian manifold to a Riemannian manifold is harmonic if the \(k^{\text{th}}\) covariant differential of its tension field vanishes. In this paper, the authors generalize this result to integral inequalities involving divergence and Laplacian of the tension field which in turn also provides a proof of the above claim (see Theorem 3.1 and Remark 3.11). For this, the authors define divergence of a vector field along a map and prove a divergence theorem for a vector field along a map, called the generalized divergence theorem (Theorem 2.2), which plays the crucial role in obtaining the mentioned generalization of the result above.