Tube surfaces with type-2 Bishop frame of Weingarten types in \(E^3\) (Q2873049)
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scientific article; zbMATH DE number 6249401
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Tube surfaces with type-2 Bishop frame of Weingarten types in \(E^3\) |
scientific article; zbMATH DE number 6249401 |
Statements
23 January 2014
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tube surfaces
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Weingarten property
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type-2 Bishop frame
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mean and Gaussian curvatures
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second Gaussian curvature
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0.9060252
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0.89934134
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0.8819979
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0.86573917
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0.86229223
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0.85646087
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Tube surfaces with type-2 Bishop frame of Weingarten types in \(E^3\) (English)
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Theorem 1 of this paper states that the tube of radius \(r\)NEWLINEaround a curve in 3-spaceNEWLINEis a Weingarten surface. It is well known that \(H\) and \(K\) satisfyNEWLINEthe linear relation \(2r H - r^2 K = 1\).NEWLINETheorems 2, 3 and 6 assume that the second fundamental form of a surfaceNEWLINEis non-degenerateNEWLINEand conclude that the surface is intrinsically flat. This is impossible.NEWLINETheorem 5 deals with the tube around a curve that satisfyNEWLINEa linear relation between \(H\) and \(K\).NEWLINEThe conclusion is that \(H\) and \(K\) both are constant,NEWLINEin contrast with the general equation above.
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