The curvature and torsion of curves in a surface
Let f be a function with certain properties and γ be a closed curve with the torsion τ . We prove that ∮ γ f τ d s = 0 if γ is a spherical curve, and conversely, if a surface makes the integral equal to zero for all closed curves, it is part of a sphere or a plane. This generalizes a known theorem o...
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Veröffentlicht in: | Journal of geometry 2017-12, Vol.108 (3), p.1085-1090 |
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Hauptverfasser: | , |
Format: | Artikel |
Sprache: | eng |
Schlagworte: | |
Online-Zugang: | Volltext |
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Zusammenfassung: | Let
f
be a function with certain properties and
γ
be a closed curve with the torsion
τ
. We prove that
∮
γ
f
τ
d
s
=
0
if
γ
is a spherical curve, and conversely, if a surface makes the integral equal to zero for all closed curves, it is part of a sphere or a plane. This generalizes a known theorem on the total torsion for a closed curve. |
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ISSN: | 0047-2468 1420-8997 |
DOI: | 10.1007/s00022-017-0397-8 |