The title is a quote from p.256 of Wilhelm Klingenberg's 1995 Riemannian Geometry (Google Books link): Every surface homeomorphic to a sphere $\mathbb{S}^2$ has three distinct, simple, closed geodesics. Another source is p.466 of Berger's A Panoramic view of Riemannian Geometry (Springer link).

My question is:

What is known about simple, closed curves of constant, non-zero geodesic curvature? Are there always three, simple closed curves for every constant $k_g$, on a surface homeomorphic to $\mathbb{S}^2$ ?

Update. macbeth noted that this question was posed on MO earlier and adequately answered: "Curves of constant curvature on S^2."

  • $\begingroup$ Ian Agol gave a link, Rugang Ye had relevant articles in 1991 and 1996 in the Pacific Journal of Mathematics. If the surface has a point $p$ which is a non-degenerate critical point of the curvature function, there is a foliation of a neighborhood of $p$ by concentric circles of constant geodesic curvature. So then you have infinitely many simple closed curves of constant but nonzero geodesic curvature. Now I've got to think about critical points of curvature. $\endgroup$
    – Will Jagy
    Apr 16, 2012 at 6:14
  • $\begingroup$ The link to which Will refers is now deleted, so here it is: "Foliation by constant mean curvature spheres," ams.org/mathscinet-getitem?mr=1084717 . $\endgroup$ Apr 16, 2012 at 12:09
  • $\begingroup$ Is this the same question as mathoverflow.net/questions/31920/… ? (Joseph O'Rourke and Will Jagy, I'm second-guessing myself because you both also commented on that question!) $\endgroup$
    – macbeth
    Apr 16, 2012 at 13:41
  • 1
    $\begingroup$ It's slightly different, since you're asking for 3, but Dmitri was asking for 2 :) $\endgroup$
    – Ian Agol
    Apr 16, 2012 at 14:46
  • 1
    $\begingroup$ I never noticed you wanted the same constant curvature $k_g.$ In that case, the number is definitely finite and likely small. $\endgroup$
    – Will Jagy
    Apr 16, 2012 at 19:36


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