Abstract
We show that the total curvature of a knotted curve is greater or equal 4π in a Hadamard Manifold. The main idea is to find a line, which intersects the knot in at least four points in a certain sequence. Although the existence of such a line is not shown, we can show the existence of lines, which are arbitrarily close to such a line in a certain sense. Finally, this yields the desired result.
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Schmitz, C. The Theorem of Fáry and Milnor for Hadamard Manifolds. Geometriae Dedicata 71, 83–90 (1998). https://doi.org/10.1023/A:1005090710096
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DOI: https://doi.org/10.1023/A:1005090710096