# Proof of a theorem of Jean-Pierre Serre on geodesics of closed Riemannian manifolds

An oft-cited 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?

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This question is not even a real question. I might as well ask you what my favourite colour is. – Yemon Choi Jun 30 '13 at 16:55
Serre has a lot of well-known theorems. This is like asking about the well-known song of Michael Jackson. – Ryan Budney Jun 30 '13 at 17:01
@Ryan, I'd say "Billy Jean". – Włodzimierz Holsztyński Jun 30 '13 at 22:13
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...." – Barry Cipra Jul 1 '13 at 19:12
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/mathscinet-getitem?mr=45386 . – David Speyer Jul 1 '13 at 20:04

The question having been reopened, I'll re-post 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...."

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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?!? – Jan Jitse Venselaar Jul 3 '13 at 16:51
@Jan, thank you. I fixed it, I think. – Barry Cipra Jul 3 '13 at 20:12