Principal, Asymptotic, and Geodesic Curves

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Homework Statement


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


Homework 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.


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.
 

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