Thus I am given $\exp_p: T_pM \to M$, where $M$ is complete and of nonpositive sectional curvature. Can one prove that if one endows $T_pM$ with the Euclidean metric that this map is an expanding map?
• An expanding map $\phi$ from a manifold $(M,g)$ to a manifold $(N,h)$ is one such that $h(\phi_*v,\phi_*v) \geq g(v,v)$ for any tangent vectors $v \in T_pM$. FYI, for the fact above about covering spaces, one needs the dimension of $M$ and $N$ to be the same but that's no problem here. – Hari Rau-Murthy Mar 25 '18 at 20:25