Let $(M,g)$ be a connected compact Riemannian manifold without boundary, let $\operatorname{Ric}^M\ge 0$, and let $(N,h)$ have $K_N\le 0$. If $u:M\to N$ is smooth and harmonic, and if the energy density $e(u):M\to\mathbb{R}$ is defined by $e(u)(p)=\frac{1}{2}|du_p|_{g,h}^2$ for $p\in M$, then $e(u)$ is constant and $\nabla du=0$.