# Curves of constant curvature on S^2

Most probably this is a well known question. Consider $S^2$ with a Riemannian metric. I would like to ask what is known about the structure of the set of simple (without self-intersections) closed curves on it of constant geodesic curvature.

Here is a series of questions.

1) Is this true that through each point of $S^2$ passes a simple closed curve of constant curvature? If not, can one estimate from below the proportion of the area of $S^2$ covered by such curves?

2) Is it true that for each value of curvature there are at least $2$ simple closed curves on $S^2$ of this curvature? Or maybe even more than $2$?

3) What can be said about the global structure of these curves on a generic $S^2$? Taking the union of all such closed curves we could try to cook up from them a surface (that maps naturally to $S^2$). Is something known about the topology of this surface?

1) The theorem of Birkhoff states that each Riemannian $S^2$ contains at least three simple closed geodesics, as Joseph remarks below.

2) For a generic metric on $S^2$ the set of such curves this set should be one dimensional. Indeed for each fixed value of curvature you can consider an analogue of the geodesic flow on the space of unite tangent vectors to $S^2$ and you expect that closed orbits will be isolated.

ADDED. Is seems indeed that these are open (and I guess hard) questions. Macbeth gave a very nice reference, that tells in particular that similar questions were raised previously by Arnold, I copy the Macbeth's reference here, so it is visible to everyone: http://count.ucsc.edu/~ginzburg/ARNOLD/mag-post.pdf

Update. The following reference : http://arxiv.org/abs/0903.1128 gives a positive answer to question 2) for spheres of non-negative Gaussian curvature provided we consider not only simple curves on $S^2$ but also curves that bound immersed disks.

One more update. There is a new nice article http://arxiv.org/abs/1105.1609 that provides some further results concerning question 2) for $S^2$ of positive curvature. This article also gives all necessary references from which one can conclude that question 2) was considered by Poincare in 1905, as it is written in the article of S.P. Novikov http://iopscience.iop.org/0036-0279/37/5/R01/pdf/0036-0279_37_5_R01.pdf

• Just a minor modification: We know that each $S^2$ contains three distinct simple closed geodesics, and this is best possible (achieved by the ellipsoid). Lyusternik and Schnirelmann, 1929. Jul 14, 2010 at 23:26
• May I ask that you clarify what you mean by the curvature of a curve? I assume you mean the geodesic curvature $k_g$, where the curvature at a point $p$ is the curvature of the curve projected onto the tangent plane at $p$? Jul 15, 2010 at 0:00
• See if this fits a comment window. Dmitri is allowing spheres that may not have an isometric embedding in $\mathbb R^3,$ although some $\mathbb R^n$ by Nash. So I suggest the term that comes up in first variation of arc length, $$\parallel \bigtriangledown_{c'}c' \parallel$$ where $c$ is a unit speed curve in your abstract surface with Riemannian metric. It is a good bet that this agrees when in the more concrete setting. From "Comparison Theorems in Riemannian Geometry" by Jeff Cheeger and David Ebin. Also en.wikipedia.org/wiki/Nash_embedding_theorem Jul 15, 2010 at 1:15
• How about the corresponding infinitisemal problem (a perturbation of the standard metric)? Jul 15, 2010 at 15:55
• I've learned a little about "magnetic flows" from Gabriel Paternain. Is it true that, on a surface, your constant-curvature flows are the magnetic flows for which the magnetic field is a constant multiple of the area form? Here's a survey by Ginzburg of some work in this area: count.ucsc.edu/~ginzburg/ARNOLD/mag-post.pdf Jul 16, 2010 at 20:54

In a recent paper, Sun proved that

i)such curves concentrate around the critical point of the Gaussian curvature

ii) there exits a curve with constant geodesic curvature in every neighborhood of a non-degenerate critical of the Gaussian curvature

My intuition is that we have

ii') there exits a foliation of a neighborhood of non-degenerate critical of the Gaussian curvature foliated by curves with constant geodesic curvature and this foliation is unique

Since we have such a result for surface with constant mean curvature, see Ye91 and Ye96.

So this gives a picture of the asymptotic structure of this moduli space as a one dimensional manifold. However i guess that that the question of the global structure is quite open.

Matthias Schneider has a nice treatment of the problem:

http://arxiv.org/abs/0808.4038

Grossly speaking, generically the space if such curves is finite, and the number, counted algebraically, is equal to 2. If the ambient space is 1/4 pinched, then the curves are simple - they do not self intersect.

• Graham hi! Thanks for the link, this is a good result indeed. Though this does not solve the problem as well as the other articles of Schneider that I cite in the question... Jun 23, 2011 at 16:41

This is an update on new results (and old conjectures) on closed curves of constant curvature. I've just spotted a new paper on arxiv on immersed curves of constant curvature. The authors prove the existence of immersed closed curves of constant geodesic curvature in an arbitrary Riemannian $$2$$-sphere for almost every prescribed curvature.

https://arxiv.org/pdf/2106.12374.pdf

There is a nice short survey in this paper about the history of the topic, on page 2. In particular, there is a reference to the following paper of Ketover and Liokumovich

https://arxiv.org/pdf/1810.09308.pdf

which attributes the following conjecture to Novikov:

Conjecture (Novikov). Every Riemannian two-sphere contains a smoothly embedded curve of curvature c for any $$0 < c < \infty$$.

The authors say that this conjecture is from Section 5 of the following Novikov's paper:

http://www.mi-ras.ru/~snovikov/74.pdf

I was not able to spot this conjecture in Novikov's paper immediately, but hopefully it's there. So it looks like part 2 of the original question is indeed an old open problem.

• Perhaps it's worth pointing out, as the authors Cheng and Zhou claim, that apparently the result also follows from work of Asselle and Benedetti, available at arxiv.org/pdf/1412.0531.pdf. Jun 24, 2021 at 17:04
• Yes, thanks for adding this comment! Jun 24, 2021 at 17:15