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The Milnor-Totaro theorem for space polygons. (English) Zbl 0990.53076

Let \(X\) be a closed generic oriented polygonal space curve in the Euclidean 3-space, and let \(k\) and \(\tau\) denote the curvature and the torsion of \(X\), respectively. The author proves that if \(\tau\) is positive, then \[ \int_X \sqrt{k^2+ \tau^2} ds>4\pi. \] Furthermore, this integral is greater or equal to \(2\pi n\) whenever no four consecutive vertices lie in a plane, and \(X\) has linking number \(n\) with a straight line. These results extend theorems of J. Milnor [Math. Scand. 1, 289-296 (1953; Zbl 0052.38402)] and B. Totaro [Int. J. Math. 1, 109-117 (1990; Zbl 0702.53004)].

MSC:

53C65 Integral geometry
53A04 Curves in Euclidean and related spaces
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