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Zapiski Nauchnykh Seminarov POMI, 1997, Volume 246, Pages 66–83
(Mi znsl549)
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This article is cited in 4 scientific papers (total in 4 papers)
Some bendings of the long cylinder
V. A. Zalgaller St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences
Abstract:
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.
Received: 24.07.1996
Citation:
V. A. Zalgaller, “Some bendings of the long cylinder”, Geometry and topology. Part 2, Zap. Nauchn. Sem. POMI, 246, POMI, St. Petersburg, 1997, 66–83; J. Math. Sci. (New York), 100:3 (2000), 2228–2238
Linking options:
https://www.mathnet.ru/eng/znsl549 https://www.mathnet.ru/eng/znsl/v246/p66
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Abstract page: | 377 | Full-text PDF : | 181 |
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