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
Hauptverfasser: Yin, Songting, Zheng, Daxiao
Format: Artikel
Sprache:eng
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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.
ISSN:0047-2468
1420-8997
DOI:10.1007/s00022-017-0397-8