Let $(M,g)$ be a compact connected smooth Riemannian manifold without boundary, and let $(N,h)$ be a complete simply connected smooth Riemannian manifold whose sectional curvature satisfies $K_N \leq 0$. Let $u_0,u_1: M \to N$ be smooth harmonic maps. For each $p \in M$, let $\gamma_p: [0,1] \to N$ be the unique constant-speed geodesic with $\gamma_p(0)=u_0(p)$ and $\gamma_p(1)=u_1(p)$, and define $H: M \times [0,1] \to N$ by $H(p,t)=\gamma_p(t)$. Then $H$ is a smooth geodesic homotopy from $u_0$ to $u_1$, and the energy

\begin{align*} E(H_t)=\frac{1}{2}\int_M |dH_t|_{g,h}^2\, d\operatorname{vol}_g(p), \qquad H_t(p):=H(p,t), \end{align*}

is independent of $t \in [0,1]$.

Moreover, if $K_N<0$ and $H(M\times[0,1])$ is not contained in the image of a single geodesic in $N$, then $u_0=u_1$.