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Principal, Asymptotic, and Geodesic Curves

  1. Apr 9, 2010 #1
    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.
  2. jcsd
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