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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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up vote 6 down vote accepted

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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I have some understanding of what you said. Can you speak more explicitly? Thanks! –  user34669 Apr 20 at 12:36
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I added a very explicit picture. –  Alexandre Eremenko Apr 20 at 12:59
    
It's so impressive! –  user34669 Apr 20 at 13:06
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