Skew loops and quadric surfaces
  A skew loop is a closed curve without parallel tangent lines. We prove:   The only complete surfaces in euclidean 3-space with a point of positive curvature and no skew loops are the quadrics.   In particular, ellipsoids are the only closed surfaces without skew loops.   We also prove results about skew loops on cylinders and positively curved surfaces. 
Mohammad Ghomi and Bruce Solomon