If there exist two geodesics from $p$ to $q$ that are not only different from each other but also infinitesimally close to each other, then it implies that $q$ is conjugate to $p$.

Can anyone give an example that $p$ is conjugate to $q$ but there don't exist two different geodesics from $p$ to $q$ that are infinitesimally close to each other?


Start with a sphere. Draw several meridians from the S pole to N pole. Then distort the metric (by growing some mountains) in the regions between these meridians.


  • $\begingroup$ I have some understanding of what you said. Can you speak more explicitly? Thanks! $\endgroup$ – 346699 Apr 20 '14 at 12:36
  • 2
    $\begingroup$ I added a very explicit picture. $\endgroup$ – Alexandre Eremenko Apr 20 '14 at 12:59
  • $\begingroup$ It's so impressive! $\endgroup$ – 346699 Apr 20 '14 at 13:06

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