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.

  • $\begingroup$ What is an expanding map? $\endgroup$ – user99914 Mar 25 '18 at 20:22
  • $\begingroup$ 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. $\endgroup$ – Hari Rau-Murthy Mar 25 '18 at 20:25

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