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?


1 Answer 1


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, 2014 at 12:36
  • 2
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.