manuscripta mathematica

, Volume 57, Issue 1, pp 101–108 | Cite as

On the total absolute curvature of closed curves in spheres

  • Eberhard Teufel


The total absolute curvature of a closed curve in a Euclidean space is always greater or equal to 2 (Fenchel's inequality,1929, [3], [1], [13]); especially for a knotted curve it is always greater than 4 (Fary-Milnor's inequality, 1949, [2], [7], [5], [4]).

For the total absolute curvature of closed curves in spheres no such lower bounds exist because there are closed geodesies. Here we derive similar bounds which depend on the length of the curve resp.the area of surfaces of disk-type bounded by the curve.

In order to prove these inequalities we start from the computation of the total absolute curvature as mean value of the number of critical points of certain level functions ([11],[12]); we use some topological considerations and Poincaré's integralgeometric formula for the computation of length resp. area.


Lower Bound Euclidean Space Number Theory Algebraic Geometry Topological Group 
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Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Eberhard Teufel
    • 1
  1. 1.Mathematisches Institut B der UniversitätStuttgart-80Federal Republic of Germany

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