Let $M=\mathfrak{R}^2-{0}$ be a manifold equipped with the metric \begin{equation} g=\frac{\langle,\rangle}{x^2+y^2}, \end{equation} where $\langle,\rangle$ is the standard Euclidean metric. Let furthermore $\Gamma=\langle \mu \rangle$ be the group generated by $\mu:(x,y)\rightarrow (2x,2y)$. Then $\Gamma$ will act properly discontinuously on $(M,g)$. Furthermore the quotient $M/\Gamma$ will be identified with the torus $S^1\times S^1$, which is compact and by standard theorems there exists a covering map $\phi:M\rightarrow M/\Gamma$.

Now on the one hand any geodesic that points towards the origin will not be complete, but on the other hand $M/\Gamma$ is compact and should be complete by Hopf-Rinows theorem.

This seems to be a contradiction. Where am I going wrong?

  • $\begingroup$ Why $\mu$ is properly discontinuous ? $\endgroup$
    – HK Lee
    Mar 8 '16 at 1:15

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