Prescribe a map:

$$\Psi:\zeta^2 \to \Bbb T^2$$

which gives a transformation of $\zeta-$space to the flat torus.

Let $\zeta^2$ consist of flow lines in which the sources correspond to $(0,0)$ and $(1,0)$ and the sinks correspond to $(0,1)$ and $(1,1)$:

$$ \tau:= \{ (x, y) \in \Bbb R^2 | \ln(x)\ln(y)=\ln^2(s) \}$$ $$ \mu:= \{ (x, y) \in \Bbb R^2 | \ln(1-x)\ln(y)=\ln^2(s) \} $$

$x,y\in \Re(0,1)$ and $0< \Re(s)<1.$

Combine the two sets of flow lines by adding their tangent spaces at each point.

I want to prove that for a stable, or rigid flow, there are infinitely many closed orbits on the flat torus after the map $\Psi:\zeta^2 \to \Bbb T^2,$ which sends vertical flow lines in $\zeta^2$ to closed orbits on $\Bbb T^2.$

A more natural site: Lorentzian geometry.

Notice that $\tau$ and $\mu$ individually are Killing flow lines preserving the same flat Lorentzian metric, $g=\frac{dxdt}{xt},$ so technically when I wrote $\Bbb R^2$ it didn't align with the natural setting for the problem, because that's a Riemannian manifold.

When these flows transversely intersect we can add up all pairs of vectors and generate a resultant Killing flow.

This suggests that the $\Bbb T^2$ we are considering must admit a Lorentzian metric and Killing field, as opposed to considering the Riemannian flat torus.

I think this restructuring of the problem is good and I didn't see it before. It remains to be seen how to show that the Killing field on the Lorentzian torus has infinitely many closed orbits.

Note: I think the construction above is sort of clunky. I think there is a more natural and elegant construction than the above, via conformal compactification of Lorentz-Minkowski, next adding pairs of vectors (space-like $+$ time-like) to get a resultant Killing flow and then stretching into the torus. This would be an alternative approach.

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    $\begingroup$ I guess you’ll have more chances to obtain an answer if your state your question in a form allowing it to be understood by a mathematician with a general background, but not only to a specialist, especially, familiar with the used notations. $\endgroup$ Apr 10, 2020 at 4:02
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    $\begingroup$ I'm working on the translation but it will take time $\endgroup$
    – geocalc33
    Apr 15, 2020 at 17:29
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    $\begingroup$ OK. Meantime, if you wish, I can try to save your bounty points by providing a fake answer. If you’ll award it with the bounty then I’ll return your points by awarding any your answer at MSE. $\endgroup$ Apr 17, 2020 at 12:07
  • $\begingroup$ Now you can choose your answer which you want to be awarded. $\endgroup$ Apr 17, 2020 at 15:17
  • $\begingroup$ I started the bounty. According to the rules, I'll may award it in 23 hours. $\endgroup$ Apr 17, 2020 at 15:30


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