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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?

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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.

http://thegraphicsfairy.com/vintage-halloween-clip-art-cute-little-pumpkin/

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  • $\begingroup$ I have some understanding of what you said. Can you speak more explicitly? Thanks! $\endgroup$
    – 346699
    Apr 20, 2014 at 12:36
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    $\begingroup$ I added a very explicit picture. $\endgroup$ Apr 20, 2014 at 12:59
  • $\begingroup$ It's so impressive! $\endgroup$
    – 346699
    Apr 20, 2014 at 13:06

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