Let $(M,g)$ be a compact smooth Riemannian manifold without boundary and let $(N,h)$ have sectional curvature $K_N\le 0$. Let $U:M\times[0,1]\to N$ be a smooth geodesic homotopy, meaning that for each $p\in M$ the path $t\mapsto U(p,t)$ is a geodesic in $N$, and write $u_t=U(\cdot,t)$. Then the function