On infinite curves on the Klein bottle

D. V. Anosov

Sbornik. Mathematics, Tome 66 (1990) no. 1, pp. 41-58

Résumé

The author investigates continuous nonselfintersecting (semi-) infinite curves $L=\{z(t);t\geqslant0\}$ on the Klein bottle $\mathbf R^2/\Gamma$, where the group $\Gamma$ of covering transformations is generated by translations through elements of the integral lattice together with the transformation $(x,y)\mapsto(x+\frac12,-y)$. It is proved that if $\widetilde L=\{\widetilde z(t)\}\subset\mathbf R^2$ is a curve which covers $L$ and goes to infinity, then $\widetilde L$ has a horizontal or vertical asymptotic direction $\widetilde l$ at infinity; that is, a ray starting at a fixed point of $\mathbf R^2$ and passing through $\widetilde z(t)$ has a horizontal or vertical limit as $t\to\infty$. In the first case (when $\widetilde l$ is horizontal) the divergence of $\widetilde L$ from $\widetilde l$ is bounded, but in the second case it can be unbounded on one side (but not on both). In passing, a simplified description is given of an example (published earlier in Trudy Mat. Inst. Steklov. 185 (1988), 30–35) demonstrating the existence of the analogous phenomenon of unbounded divergence for the torus. Bibliography: 8 titles.