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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].

57R30 Foliations in differential topology; geometric theory