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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.