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?



Question about conjugate points
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/> 


