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On the lifts to the plane of semileaves of foliations on the torus with a finite number of singularities. (English. Russian original) Zbl 0972.57018
Proc. Steklov Inst. Math. 224, 20-45 (1999); translation from Tr. Mat. Inst. Steklova 224, 28-55 (1999).
A curve without selfcrossings $$\varphi:\mathbb{R}_+\to\mathbb{T}^2$$ is constructed in the two-dimensional torus $$\mathbb{T}^2$$ with the property that for every foliation on $$\mathbb{T}^2$$ with a finite number of singularities, for every semileaf $$F$$, for all lifts $$\overline\varphi :\mathbb{R}_+ \to\mathbb{R}^2$$ and $$\overline F\subset\mathbb{R}^2$$ of $$\varphi$$ and $$F$$, the curves $$\overline\varphi (\mathbb{R}_+)$$ and $$\overline F$$ are on infinite Fréchet distance. The latter means that there are a constant $$c$$ and injective parametrizations $$z:\mathbb{R}_+\to\mathbb{R}^2$$ and $$w:\mathbb{R}_+ \to\overline F$$ of the curves $$\overline\varphi (\mathbb{R}_+)$$ and $$\overline F$$ such that $$\|z(t)- w(t)\|\leq c$$ for all $$t\in \mathbb{R}_+$$.
For the entire collection see [Zbl 0942.00074].

##### MSC:
 57R30 Foliations in differential topology; geometric theory
Fréchet distance