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Abstract.A skew loop is a closed curve without parallel tangent lines. We prove: The only complete surfaces in R 3 with a point of positive curvature and no skew loops are the quadrics. In particular: Ellipsoids are the only closed surfaces without skew loops. Our efforts also yield results about skew loops on cylinders and positively curved surfaces. Mathematics Subject Classification (2000). Primary 53A04, 53A05; Secondary 53C45, 52A15.

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