So I was told in a class that for a complete Riemannian manifold, $M$, with non-positive sectional curvature the exponential map at any point $p\in M$ is a covering map. Part of the proof just assumed that the exponential map is surjective, but why is it so? I know that since our manifold is complete $exp_p$ is defined on all of $T_pM$, but I don't see why it is surjective. Can someone help me out. Thanks.

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    $\begingroup$ The surjectivity of exp. for complete manifolds is known as the Hopf-Rinow Theorem. Milnor's book on Morse Theory contains an elegant proof. $\endgroup$ – Thomas Nov 18 '15 at 17:58

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