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Prescribe a map:

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

which gives a transformation of $\zeta$-space to the "square" flat torus, by identifying $(x,y)\sim (x+1,y)\sim(x,y+1).$

Let $(\zeta^2,g)$ with $g=\frac{dxdy}{xy}$ for $x,y \in (0,1).$ Then we have a Cauchy foliation of $\zeta^2$ defined by:

$$ \mathcal{F_s}=\big\lbrace \log x \log y=s: s>0\big\rbrace $$

and a flow tangent to $\mathcal {F}_s$ which is Killing.

Is it possible to impose conditions s.t. the image of $\mathcal{F}_s$ under $\Psi$ is also Killing?

John Zimmerman
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    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. – Alex Ravsky Apr 10 '20 at 04:02
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    I'm working on the translation but it will take time – John Zimmerman Apr 15 '20 at 17:29
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    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. – Alex Ravsky Apr 17 '20 at 12:07
  • Now you can choose your answer which you want to be awarded. – Alex Ravsky Apr 17 '20 at 15:17
  • I started the bounty. According to the rules, I'll may award it in 23 hours. – Alex Ravsky Apr 17 '20 at 15:30

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