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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
DOI: https://doi.org/10.1090/S0002-9939-04-07310-1
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, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1989. MR 92h:53003
  • 2. S. Lang, Analysis I, Addison-Wesley Publishing Company, Reading, MA, 1969.
  • 3. J. W. Milnor, On the total curvature of knots, Ann. of Math. 52, No. 2 (1950), 248-257. MR 12:273c
  • 4. J. A. Wolf, Spaces of Constant Curvature, McGraw-Hill, Inc., NY, 1967. MR 36:829

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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: https://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

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