I met a problem to prove or disprove that: a. If a curve is both an asymptotic curve and a line of curvature, then it must be planar. b. If a curve is planar and an asymptotic curve, then it must be a line.
For a, I show that the derivative of the shape operator along the curve is 0 and concluding that it is true, although I am not fully convinced by my argument. However, for b, I am totally off.
This question is different from Geodesics and Curves on a Plane. The condition here is asymptotic, line of curvature and planar, but not geodesic. Please carefully reading the questions.