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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8$\begingroup$ This question is not even a real question. I might as well ask you what my favourite colour is. $\endgroup$– Yemon ChoiCommented Jun 30, 2013 at 16:55
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11$\begingroup$ Serre has a lot of well-known theorems. This is like asking about the well-known song of Michael Jackson. $\endgroup$– Ryan BudneyCommented Jun 30, 2013 at 17:01
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5$\begingroup$ @Ryan, I'd say "Billy Jean". $\endgroup$– Włodzimierz HolsztyńskiCommented Jun 30, 2013 at 22:13
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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 CipraCommented Jul 1, 2013 at 19:12
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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/mathscinet-getitem?mr=45386 . $\endgroup$– David E SpeyerCommented Jul 1, 2013 at 20:04
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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...."
answered Jul 1, 2013 at 23:14