I was wondering whether it is possible to prove the Cartan Hadamard theorem by using the fact that every expanding map is a covering map.
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?
I'm interested because this is the first thought that came into my head when I tried to prove this theorem myself.