1. The problem statement, all variables and given/known data Is there a curve on a regular surface M that is asymptotic but not principal or geodesic? 2. Relevant equations The given definitions of asymptotic, principal, and geodesic: A principal curve is a curve that is always in a principal direction. An asymptotic curve is a curve [tex]\alpha[/tex] where [tex]\alpha ''[/tex] is tangent to M. A geodesic curve is a curve [tex]\alpha[/tex] where [tex]\alpha ''[/tex] is normal to M. 3. The attempt at a solution The closest I have come is finding a curve that is both asymptotic and geodesic, where the example comes on the saddle surface [tex]z=xy[/tex] and the curves would be the x and y axes. I just can't think of a surface where there is only an asymptotic curve. This is not a homework assignment, more of a question I have been asking to try to understand the material better.