Let $\phi : M \to \mathbb{R}^n$ be a harmonic map, where $M$ is a Riemannian manifold. Let us take coordinates $(u_1, u_2,..., u_n)$ on $\mathbb{R}^n$ and express the Euclidean metric as $g = \Sigma g_{ij}du^idu^j$. Write now the symmetric bilinear form $\phi^*(g)$ on $M$.
My question is, is there a characterization of the fact that $\phi$ is harmonic in terms of the symmetric bilinear form $\phi^*(g)$? In other words, is there a necessary and/or sufficient condition on $\phi^*(g)$ such that $\phi$ is a harmonic map? Thanks!