Abstract
Let f be a function with certain properties and \(\gamma \) be a closed curve with the torsion \(\tau \). We prove that \(\oint _{\gamma }f\tau ds=0\) if \(\gamma \) 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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chen, Y.F.: A new proof of a theorem of total torsion and its generalization. J. Fujian Norm. Univ. 3, 11–14 (1987) (in Chinese)
Geppert, H.: Sopra una caratterizzazione della sfera. Ann. Mat. 4, 59–66 (1941)
Pansonato, C.C., Costa, S.I.R.: Total torsion of curves in three-dimensional manifolds. Geom. Dedic. 136, 111–121 (2008)
Qin, Y.A., Li, S.J.: Total torsion of closed lines of curvature. Bull. Aust. Math. Soc. 65, 73–78 (2002)
Santaló, L.A.: Algunas propriedades de las curvas esféricas y una característica de la esfera. Rev. Mat. Hispanoam. 10, 1–4 (1935)
Segre, B.: Atti della Acad. Nazi. dei Lincei. Rendiconti 3, 420–422 (1947)
Shen, Y.B.: Global differential geometry. Higher education Press, Beijing (2009) (in Chinese)
Wang, Y.N.: Remarks on spherical curves in \(R^3\). J. Beijing Norm. Univ. 29(3), 304–307 (1993) (in Chinese)
Author information
Authors and Affiliations
Corresponding author
Additional information
This project is supported by AHNSF (1608085MA03) and TLXYRC(2015tlxyrc09).
Rights and permissions
About this article
Cite this article
Yin, S., Zheng, D. The curvature and torsion of curves in a surface. J. Geom. 108, 1085–1090 (2017). https://doi.org/10.1007/s00022-017-0397-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00022-017-0397-8