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On the total absolute curvature of closed curves in spheres

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Abstract

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.

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  1. Mathematisches Institut B der Universität, Pfaffenwaldring 57, D-7000, Stuttgart-80, Federal Republic of Germany

    Eberhard Teufel

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  1. Eberhard Teufel
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Teufel, E. On the total absolute curvature of closed curves in spheres. Manuscripta Math 57, 101–108 (1986). https://doi.org/10.1007/BF01172493

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  • DOI: https://doi.org/10.1007/BF01172493

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