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Total torsion of closed lines of curvature

Published online by Cambridge University Press:  17 April 2009

Yong-An Qin  and
Shi-Jie Li
Yong-An Qin
Affiliation:
Department of Mathematics, South China University of Technology, Guangzhou 510641, China e-mail: [email protected]
Shi-Jie Li
Affiliation:
Department of Mathematics, South China Normal University, Guangzhou 510631China e-mail: [email protected]
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Abstract

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In this article we investigate the total torsion of closed lines of curvature on a surface in 3 and obtain the following results.

(1) The total torsion of a closed line of curvature on a surface is kπ where k is an integer. Conversely, if the total torsion of a closed curve is kπ for an integer k, then the curve can appear as a line of curvature on a surface. In particular, if the total torsion of a closed curve is 2kπ, then it can appear as a line of curvature on a closed, oriented surface of genus 1.

(2) The total torsion of a closed line of curvature on an ovaloid is zero.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 65 , Issue 1 , February 2002 , pp. 73 - 78
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Blaschke, W., Kreis und Kugel (Wlater de Gruyter and Co., Berlin, 1956).CrossRefGoogle Scholar
[2]Chen, Y.F., ‘A new proof of a theorem of total torsion and its generalization’, (in Chinese), Fujian Shifan Daxue Xuebao Ziran Kexue Ban 3 (1987), 11–14.Google Scholar
[3]Geppert, H., ‘Sopra una caratterzione della spera’, Ann. Mat. Pura. Appl. 20 (1941), 59–66.CrossRefGoogle Scholar
[4]Millman, S. and Parker, D., Elements of differential geometry (Prentice-Hall, Inc., New Jersey, 1977).Google Scholar
[5]Spivak, M., A comprehensive introduction to differential geometry, Vol. III (Publish or Perish Inc., Wilmington, Del., 1979).Google Scholar