Is there a curve on a regular surface M that is asymptotic but not principal or geodesic?
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.
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.