Journal of Mathematical Sciences

, Volume 100, Issue 3, pp 2228-2238

First online:

Some bendings of a long cylinder

  • V. A. Zalgaller

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Elementary tools are applied to describe piecewise-linear isometric embeddings of cylindrical surfaces in ℝ3. Let T2 be a flat torus, let γ⊂T2 be the shortest closed geodesic of length lo, and let k be a fixed positive integer. We assume that if l is the length of any closed geodesic on T2 which is homotopic neither to γ nor to any power of γ, then l>kl0. It is shown how to embed T2 in ℝ3 if k is sufficiently large. The same problem is solved for a flat skew torus T2. It is also shown that if a knot of arbitrary type in ℝ3 is fixed and k is sufficiently large, then T2 can be isometrically embedded in ℝ3 as a tube knotted according to the type of fixed knot. Bibliography; 4 titles.