Simple, closed geodesics in $\mathbb{S}^3$ manifold

Question

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.

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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 $\mathbb{S}^2$.

Answer 1

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 $s = n - 2^{\lfloor \log_2 n\rfloor}.$

Klingenberg, Wilhelm, On the existence of closed geodesics on spherical manifolds, Math. Z. 176, 319-325 (1981). ZBL0469.53042, MR610213.

@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 = {https://doi.org/10.1007/BF01214609}, }

Answer 2

Just to set the terminology, a closed geodesic is called simple when it is a smooth embedded circle in the Riemannian or Finsler manifold. If a closed Riemannian manifold is not simply connected one can easily show that the shortest non-contractible curve is a simple closed geodesic. Existence and multiplicity results for simple closed geodesics on simply connected manifolds are currently available only for 2-dimensional manifolds. Indeed, as it has been already mentioned, in higher dimension being simple is a generic condition: given a closed geodesic $\gamma$, with a tiny perturbation of the metric one can produce a simple closed geodesic close to $\gamma$ for the new metric. For this reason, it is not obvious how to obtain a simple closed geodesic in higher dimension by means of variational methods: a generic metric has at least one (and often more), but a very unlucky metric in principle might have none.

The existence of closed geodesics possibly with self-intersections was proved by Birkhoff for Riemannian 2-spheres, and later by Lusternik-Fet in full generality (as it was mentioned above by Rivin and Alvarez Paiva). These proofs remain valid for general Finsler metrics.

Beside the already mentioned case of Riemannian $S^2$, the existence of a second closed geodesic is already hard business: for suitably non-degenerate Riemannian metrics, it was a theorem of Fet from the 1960s. The theorem should hold as well for reversible Finsler metrics (i.e. Finsler metrics $F$ with the property that $F(v)=F(-v)$ for all vectors $v$). For general, non-reversible, Finsler metrics on $S^2$ it was a theorem of Bangert-Long from the late 2000s. For higher dimensional simply connected closed manifolds equipped with non-degenerate and possibly non-reversible Finsler metric, the existence of a second geodesic is a recent theorem of Duan-Long-Wang (JDG, 2015).

For simply connected closed manifolds whose cohomology ring $H^*(M;\mathbb Q)$ is not generated by one element, a celebrated result of Gromoll-Meyer from 1971 implies that there are always infinitely many Riemannian or Finsler closed geodesics (possibly with self-intersections). The only simply connected manifolds not covered by this theorem are those who have the rational cohomology of a compact rank-one symmetric space, that is, the rational cohomology of $S^n$, $\mathbb RP^n$, $\mathbb CP^n$, $\mathbb HP^n$, or $\mathrm{Ca}P^2$. The case of $S^2$ was settled in a combination of celebrated papers by Bangert, Franks, and Hingston. For the other CROSSes, Hingston and Rademacher proved that a suitably generic Riemannian metric has infinitely many closed geodesics.

On non-simply connected closed Riemannian or Finsler manifolds, it is very easy to find infinitely many closed geodesics when, for instance, the fundamental group is abelian and has rank larger than 1. The statement is still true, but non-trivial, if the fundamental group is only assumed to be infinite abelian (the most difficult case being when it is $\mathbb Z$); it was proved by Bangert-Hingston in the 1980s.

In his book "Lectures on closed geodesics" from the 1980s, Klingenberg has a proof of the existence of infinitely many closed geodesics for any closed Riemannian manifold (of dimension at least two). However, the proof was crucially based on a divisibility lemma that was later found to be wrong. There are unfortunately other mistakes in the literature, notably by Alber (his so-called Alber Lemma) and Klingenberg. The results that I mentioned above are universally accepted and have a solid proof, but one should be careful with older results in the published literature which claim stronger conclusions.

The unconditional existence of infinitely many closed geodesics, or even of a second closed geodesic on a general closed Riemannian manifold of arbitrary dimension, are still open problems.

The above are among the most significant results in the field, but I certainly forgot to mention many others.

Answer 3

Since no one has pointed out this paper yet, I suggest looking at Theorem 4.2 and Theorem 4.5 in B. White - The Space of Minimal Submanifolds for Varying Riemannian Metrics. I report here the two theorems in the case of geodesics (i.e., $k=1$ with the notation of White's paper) in case someone doesn't have access to the papers.

Theorem 4.2.

(1) Almost every $C^3$ metric on $\mathbb{S}^n$ close to the round metric admits at least $n+1 \choose 2 $ embedded closed geodesics.

(2) Every $C^3$ metric on $\mathbb{S}^n$ close to the round metric admits at least Lusternik–Schnirelmann category of $\textrm{Gr}(n+1,2)$ (linear Grassmanian of unoriented $2$-planes in $\mathbb{R}^{n+1}$) embedded closed geodesics.

Note that what Alber showed in "Alber, On periodicity problems in the calculus of variations in the large, Uspehi. Mat. Nauk 12 (1957), 57-124 (Russian)" is that the Lusternik–Schnirelmann category of $\textrm{Gr}(n+1,2)$ is at least $n+2^{\lfloor \log_2(n) \rfloor}-1$. There should be a translated version in English of this work but I was not able to find it quickly.

Theorem 4.5. Let $\gamma_0$ be the round metric on $\mathbb{S}^n$, then there is $\epsilon>0$ with the following property. If $1\le a_1<a_2 < \dotsc <a_{n+1} \le 1+\epsilon$ and $\gamma$ is the metric on $\mathbb{S}^n$ making $E:=(\mathbb{S}^n, \gamma)$ an ellipsoid of sides $(a_1, \dotsc, a_{n+1})$ then there are exactly $n+1 \choose 2 $ embedded $\mathbb{S}^1$ in $E$ that are geodesics, and they are the ones relative to the $n+1 \choose 2 $ coordinate $2$-planes.