Infinite curves on the torus and on closed surfaces of negative Euler characteristic (Q919687)

Let \({\mathcal M}\) be a compact manifold (without boundary) of dimension two, with Euler characteristic \(\chi\leq 0\) and let p: \(\tilde {\mathcal M}\to {\mathcal M}\) be its universal covering; \(\tilde {\mathcal M}\) can be identified with \({\mathbb{R}}^ 2\) if \(\chi =0\) and with the Lobachevskii plane if \(\chi <0\). Consider a standard metric \({\tilde \rho}\) on \(\tilde {\mathcal M}\) inducing the metric \(\rho\) on \({\mathcal M}\). Consider a flow on \({\mathcal M}\) and corresponding (covering) flow on \(\tilde {\mathcal M}\). Let \(\tilde L=\{z(t):\) \(0\leq t<\infty \}\) be a positive semitrajectory in \(\tilde {\mathcal M}\) and assume that (1) lim \({\tilde \rho}\)(a,z(t))\(=0\) as \(t\to \infty\), where a is a fixed point of \(\tilde {\mathcal M}\). The condition (1) implies the following property: the ray from a to z(t) tends to some limit ray as \(t\to \infty\); this defines an asymptotic direction of \(\tilde L.\) The following question is natural: is it true that if \(\tilde L\) is an unbounded continuous curve in \(\tilde {\mathcal M}\) such that (1) is satisfied and \(L=p(\tilde L)\subset {\mathcal M}\) does not have self- intersection points, then the distance between L and any direct line having the asymptotic direction of \(\tilde L\) is bounded? The paper by \textit{V. I. Pupko} [Dokl. Akad. Nauk SSSR 177, 272-274 (1967; Zbl 0172.488)] gives a positive answer, but unfortunately the proof is incorrect. The aim of the paper under review is to give a counterexample by constructing a suitable (quite nontrivial) flow on a torus. There are also remarks extending the main idea for all manifolds with negative Euler characteristics.