Lyusternik and Shnirel'man were the first to prove Poincaré's conjecture that any Riemannian metric on $\mathbb{S}^2$ has at least three simple (non-self-intersecting), closed geodesics. See, e.g., p.466 of Berger's A Panoramic view of Riemannian Geometry, or this Encyc.Math. article. (Apparently details in the L.-S. 1929 proof were not resolved until 1978 1993.)

Q. Is there an extension to any Riemannian metric for $\mathbb{S}^3$? E.g., there exist at least $k > 1$ simple, closed geodesics in $\mathbb{S}^3$ under every Riemannian metric?

(Added.) As Igor Rivin points out, $k=1$ is known for any smooth metric on $\mathbb{S}^n$ via a 1927 proof of Birkoff.

Added a bounty. Much remains unclear to me, despite Igor's (extremely) useful citations. The bounty is offered for clarifying the situation (comments cited in several instances):

(1) Klingenberg proves (Mathematische Zeitschrift}) in 1981 there are 4 closed geodesics on $\mathbb{S}^3$.

(2) Long&Duan prove (Advances in Mathematics) in 2009 that there at least 2 closed geodesics on $\mathbb{S}^3$—What happened in the 28 yrs between? Was Klingenberg's proof not accepted, or did he prove something nuancedly different?

(3) And it seems that none of these authors are addressing simplicity, as @alvarezpaiva noted. The L.&S. theorem explicitly proves non-self-intersection on $\mathbb{S}^2$.

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    $\begingroup$ The non-self-intersection on $\mathbb{S}^2$ is obviously different, since self-intersection is very unlikely in higher dimensions. $\endgroup$ – Igor Rivin Sep 14 '14 at 14:48

Yes, there is at least one simple closed geodesic for every smooth metric on $\mathbb{S}^n.$ This is a result of G. D. Birckhoff (1927) - ("Dynamical Systems, AMS Coll. Pub. vol 9).

It was shown by Lyusternik that there were at least $n$ closed geodesics on $\mathbb{S}^n,$ and the sharp result (Alber-Klingenberg) is that there are $2n-s - 1,$ where $s = n - 2^{\lfloor \log_2 n\rfloor}.$

@article {MR610213, AUTHOR = {Klingenberg, Wilhelm}, TITLE = {On the existence of closed geodesics on spherical manifolds}, JOURNAL = {Math. Z.}, FJOURNAL = {Mathematische Zeitschrift}, VOLUME = {176}, YEAR = {1981}, NUMBER = {3}, PAGES = {319--325}, ISSN = {0025-5874}, CODEN = {MAZEAX}, MRCLASS = {58E10 (49F99)}, MRNUMBER = {610213 (82d:58024)}, MRREVIEWER = {Y. Mut{\^o}}, DOI = {10.1007/BF01214609}, URL = {http://dx.doi.org/10.1007/BF01214609}, }

  • $\begingroup$ Ah, Yes, my mistake for not making it clear that I knew Birkoff's result (which pre-dated L.&S.), but was looking for $k>1$. I will so edit. Thanks! $\endgroup$ – Joseph O'Rourke Sep 10 '14 at 0:19
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    $\begingroup$ @JosephO'Rourke Ok, now it's bigger than $1.$ $\endgroup$ – Igor Rivin Sep 10 '14 at 0:31
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    $\begingroup$ @JosephO'Rourke OK, here you go. $\endgroup$ – Igor Rivin Sep 10 '14 at 0:39
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    $\begingroup$ I thought that the proof for the second closed geodesic on the three-sphere was pretty recent (Long, I think). Does Klingenberg prove the existence of simple closed geodesic? Oh well, I won't be lazy and take a look at the reference. $\endgroup$ – alvarezpaiva Sep 10 '14 at 8:28
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    $\begingroup$ @alvarezpaiva: Y.Long & H.Duan, "Multiple closed geodesics on 3-spheres," Advances in Mathematics 221.6 (2009): 1757-1803. $\endgroup$ – Joseph O'Rourke Sep 10 '14 at 11:18

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