Journal of Mathematical Sciences

, Volume 100, Issue 3, pp 2228–2238

Some bendings of a long cylinder

  • V. A. Zalgaller


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.


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  1. 1.
    S. Yu. Afon'kin and E. Yu. Afon'kina, Origami: Games and Tricks with Paper [in Russian], Petersburg Origami Center, Saint Petersburg (1994).Google Scholar
  2. 2.
    C. P. Rourke and B. J. Sandersen, Introduction to Piecewise-Linear Topology (Ergeb. Math., 69), Springer, Berlin (1972).MATHGoogle Scholar
  3. 3.
    Yu. D. Burago and V. A. Zalgaller, “Isometric piecewise-linear embeddings of surfaces with polyhedral metric in ℝn,” Algebra Analiz, 7, No. 3, 76–95 (1995).MATHGoogle Scholar
  4. 4.
    A. D. Milka, “Linear bendings of regular convex polyhedra,” Mat. Fiz., Anal., Geom., 1, No. 1, 116–130 (1994).MATHGoogle Scholar

Copyright information

© Kluwer Academic/Plenum Publishers 2000

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  • V. A. Zalgaller

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