2
$\begingroup$

Let $S$ be a smooth, closed, genus zero surface in $\mathbb{R}^3$. $S$ has at least three simple (non-self-intersecting), closed geodesics by a theorem of Lyusternik and Shnirel'man. Alternatively, any Riemannian metric on $\mathbb{S}^2$ leads to at least three simple, closed geodesics.


          Ellipsoid
          (Image from Wikipedia.)


Q1. Is it true that, for every even $n > 2$, there is an $S$ that has a pair of simple, closed geodesics that cross each other $n$ times?

On an ellipsoid, each pair of the simple, closed geodesics cross twice. This feels like the generic situation. But my guess is that the answer to Q1 is Yes; but an explicit construction is eluding me.

Q2. Is there any sense in which $n=2$ crossings is the usual situation?

$\endgroup$
  • 2
    $\begingroup$ How about a case when a pair you are looking for is infinitely close close pair of geodesics (Jacobi equation)? It could be a starting point in order to construct an example… $\endgroup$ – Petya Mar 12 '15 at 14:25
3
$\begingroup$

The answer to Q1 is 'yes'. You can construct an explicit example as follows: Take a band $B$ around a geodesic $e$ (for 'equator') on the round sphere of radius 1, say, between two parallel circles above and below it. Now take the $n/2$-fold cover of this band, say, $\hat B$, and isometrically embed $\hat B$ into $\mathbb{R}^3$ as a surface of revolution and then smoothly close it up by attaching two disks, which can be done while keeping the result a genus $0$ surface $\Sigma$ of revolution. Now, the $n/2$-fold cover $\hat e$ of the geodesic $e$ will intersect all of the 'near-by' geodesics (which are also closed) exactly $n$ times. (Here, 'near-by' means geodesics that stay within $\hat B\subset\Sigma$.)

Q2 does not seem to me to be well-formulated.

$\endgroup$
  • $\begingroup$ Great, Robert! Yes, Q2 is not well-formulated. I need to investigate papers such as Jean-François Le Gall's "Random geometry on the sphere" (arXiv abs link). $\endgroup$ – Joseph O'Rourke Mar 12 '15 at 15:46
  • 6
    $\begingroup$ It seems that another example could be constructed by simply adding $n$ spikes along an equator of the sphere; the two curves that zigzag between the pikes and locally minimize length would have the required intersection. $\endgroup$ – Benoît Kloeckner Mar 12 '15 at 16:15
  • 1
    $\begingroup$ @BenoîtKloeckner: That's very simple and clever! Is it clear that those zig-zags are geodesics? $\endgroup$ – Joseph O'Rourke Mar 12 '15 at 18:27
  • 2
    $\begingroup$ @JosephO'Rourke: I think it's clear 'physically' that there will exist such geodesics if you place $n$ large bumps long the equator so that it's easier to go around them than over them. However, explicitly writing them down would be a challenge, I think. Benoît's example has the advantage that it's 'stable' in some sense because those geodesics should persist under small perturbations of the metric. On the other hand, my example produces a nonempty open set $U$ of closed geodesics on the sphere such that any two distinct geodesics in $U$ intersect in exactly $n$ points. $\endgroup$ – Robert Bryant Mar 12 '15 at 19:30
  • $\begingroup$ @BenoîtKloeckner & RobertBryant: Thanks to you both! Very insightful. $\endgroup$ – Joseph O'Rourke Mar 12 '15 at 19:39

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.