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Can you tell me, wh the following is true: If $f: M \rightarrow N$ is a smooth map between complete and connected Riemannian manifolds which fulfills $f^{*}(g_n)=g_M$ for the metrics $g_m$ and $g_N$ on $M$ and $N$, then it is a covering map?

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    The passage I was thinking of in Do Carmo is not quite as I remembered. They hypothesis is simply that $\|d_p f v\| \geq \|v\|$ for all vectors $v$ (much weaker than $f^*(g_n) = g_m$), but he also assumes $f$ is a local diffeomorphism, which is quite strong. On the other hand, being a local diffeomorphism follows from $f^*(g_n) = g_m$ together with the fact that, say, $M$ and $N$ have the same dimension (as Mariano suggested).2012-02-13

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Yes, this is true, even without the assumption that N is complete. See Lemma 11.6 in my book "Riemannian Manifolds: An Introduction to Curvature."