An oftcited theorem of Serre states that there are infinitely many geodesics between any two points in a closed Riemannian manifold. Could someone please provide an intuitive sketch of the proof?

8$\begingroup$ This question is not even a real question. I might as well ask you what my favourite colour is. $\endgroup$ – Yemon Choi Jun 30 '13 at 16:55

11$\begingroup$ Serre has a lot of wellknown theorems. This is like asking about the wellknown song of Michael Jackson. $\endgroup$ – Ryan Budney Jun 30 '13 at 17:01

5$\begingroup$ @Ryan, I'd say "Billy Jean". $\endgroup$ – Włodzimierz Holsztyński Jun 30 '13 at 22:13

4$\begingroup$ There is a paper by Nabutovsky and Rotman, available at math.toronto.edu/alex/morseoct12.pdf in which they give "a somewhat modernized sketch of the proof of Serre’s theorem given by A.Schwarz...." $\endgroup$ – Barry Cipra Jul 1 '13 at 19:12

6$\begingroup$ The current edit of the question asks a well defined mathematical question, which Barry Cipra gives a good answer to. The original paper of Serre appears to be ams.org/mathscinetgetitem?mr=45386 . $\endgroup$ – David E Speyer Jul 1 '13 at 20:04
The question having been reopened, I'll repost my comment as an answer: There is a paper by Nabutovsky and Rotman, available at http://www.math.toronto.edu/alex/morseoct12.pdf in which they give "a somewhat modernized sketch of the proof of Serre’s theorem given by A.Schwarz...."

$\begingroup$ This link doesn't work, you need to add www between http:// and math. I cannot seem to fix this: edits should at least be 6 characters?!? $\endgroup$ – Jan Jitse Venselaar Jul 3 '13 at 16:51
