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JOURNALS // Zapiski Nauchnykh Seminarov POMI // Archive

Zap. Nauchn. Sem. POMI, 1997 Volume 246, Pages 66–83 (Mi znsl549)

This article is cited 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.

UDC: 514.113, 514.752.43

Received: 24.07.1996


 English version:
Journal of Mathematical Sciences (New York), 2000, 100:3, 2228–2238

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