Some bendings of the long cylinder

V. A. Zalgaller

Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 2, Tome 246 (1997), pp. 66-83

Résumé

The peace-linear isometric embeddings of the cylindrical surfaces in $\mathbb R^3$ are described by elementary means. Let $T^2$ be a flat torus, and $\gamma$ the shortest closed geodesics on this torus of length $l_0$. Let $l$ be the length of some closed geodesics on $T^2$, which is not homotopic to $\gamma$, nor to any power of $\gamma$ and $l>kl_0$. It is demonstrated how for sufficiently large $k$ the torus $T^2$ can be embedded into $\mathbb R^3$. The same is done for the skew flat torus. For any type of knot in $\mathbb R^3$ and for sufficiently large $k$, in the isometrical embedding of the torus $T^2$ into $\mathbb R^3$ is described as a tube knotted as the above-mentioned knot.