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This is a question on Do Carmo's book "Riemannian Geometry" (question 7 from chapter 7):

Let $f:M\to \bar{M}$ be a diffeomorphism beetwen two riemannian manifolds. Suppose $\bar{M}$ complete and that there is $c>0$ such that: $$|v|\geq c|df_pv|$$ for every $p\in M$ and $v\in T_pM$. Prove that $M$ is complete.

I think one approach could be using the Hopf-Rinow Theorem, because this question is in a chapter about this theorem.Thanks.

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up vote 6 down vote accepted

Let $\{x_n\}$ be a Cauchy sequence in $M$. Lets see that $\{x_n\}$ is convergent on $M$. This shows $M$ is a complete metric space. By Hopf-Rinow's theorem, $M$ is complete, in geodesic sense.

Claim: $d_{\bar{M}}(y_m,y_n)\leqslant \frac{1}{c}d_M(x_m,x_m)$

Proof: given a curve $\gamma$ in $M$, we have $$\ell(\gamma)=\int|\gamma'|dt\geqslant c\int|df_{\gamma}\gamma'|=c\cdot \ell(f(\gamma))$$

Once $f$ is a diffeomorfism, the differentiable curves of $M$ and $\bar{M}$ are in bijection. So, considering the curves joinning $x_m$ to $x_n$ (and its images joinning $y_m$ to $y_n$), we can take infimum and it leads us to the result.

Claim: $y_n=f(x_n)$ is Cauchy in $\bar{M}$.

Proof: it follows directly from the claim above.

Hence, once $\bar{M}$ is complete, we have $y_n\rightarrow y$ in $\bar{M}$. But $f$ is a diffeomorphism. Then $x_n=f^{-1}(y_n)\rightarrow f^{-1}(y)$ in $M$.

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well written, thanks! – Jr. Jul 8 '12 at 5:06
You're welcome! – matgaio Jul 8 '12 at 5:07

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