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Total curvatures of a closed curve in Euclidean $n$-space


Authors: L. Hernández Encinas and J. Muñoz Masqué
Translated by:
Journal: Proc. Amer. Math. Soc. 132 (2004), 2127-2132
MSC (2000): Primary 53A04; Secondary 28A75, 51M20
Published electronically: January 23, 2004
MathSciNet review: 2053986
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Abstract | References | Similar Articles | Additional Information

Abstract: A classical result by J. W. Milnor states that the total curvature of a closed curve $C$ in the Euclidean $n$-space is the limit of the total curvatures of polygons inscribed in $C$. In the present paper a similar geometric interpretation is given for all total curvatures $\int_{C}\vert\kappa _{r}\vert\mathrm{d}s$, $r=1,\ldots,n-1$.


References [Enhancements On Off] (What's this?)

  • 1. A. D. Alexandrov and Yu. G. Reshetnyak, General theory of irregular curves, Mathematics and its Applications (Soviet Series), vol. 29, Kluwer Academic Publishers Group, Dordrecht, 1989. Translated from the Russian by L. Ya. Yuzina. MR 1117220
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  • 3. J. W. Milnor, On the total curvature of knots, Ann. of Math. (2) 52 (1950), 248–257. MR 0037509
  • 4. Joseph A. Wolf, Spaces of constant curvature, McGraw-Hill Book Co., New York-London-Sydney, 1967. MR 0217740

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Additional Information

L. Hernández Encinas
Affiliation: Instituto de Física Aplicada, Consejo Superior de Investigaciones Cientificas, Calle Serrano 144, 28006-Madrid, Spain

J. Muñoz Masqué
Affiliation: Instituto de Física Aplicada, Consejo Superior de Investigaciones Cientificas, Calle Serrano 144, 28006-Madrid, Spain
Email: \luis, jaime\@iec.csic.es

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07310-1
Keywords: Curvatures of a curve, Fr\'enet frame, polygon, total curvature
Received by editor(s): February 26, 2003
Received by editor(s) in revised form: March 25, 2003
Published electronically: January 23, 2004
Additional Notes: This work was supported by Ministerio de Ciencia y Tecnología (Spain) under grants TIC2001–0586 and BFM2002–00141.
Communicated by: Jon G. Wolfson
Article copyright: © Copyright 2004 American Mathematical Society