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
Schlagworte:	Curvature Geometry Mathematics Mathematics and Statistics Torsion
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description	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.
doi_str_mv	10.1007/s00022-017-0397-8
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subjects	Curvature Geometry Mathematics Mathematics and Statistics Torsion
title	The curvature and torsion of curves in a surface
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