6
$\begingroup$

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.

$\endgroup$
8
$\begingroup$

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 $M$.

$\endgroup$
  • $\begingroup$ Where was noncompactness used? This is an assumption in do Carmo. $\endgroup$ – TuoTuo Apr 20 '18 at 15:01

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.