Asymptotic Directions of a Cylinder Say I am looking at a cylinder. I have found the shape operator and I have found the eigenvalues to be k1 = -1/a and k2=0. I have also found the principal directions 
{1,0} and {0,1}. I know that if k1k2 <= 0 then there should be asymptotic directions. How can I find the asymptotic directions? How can I identify the lines of curvature and asymptotic curves? 
Note: The cylinder is parameterized by: X(u,v) = (acosu, asinu, v). 

Note that because the coordinate directions are principal directions, the coordinate curves are lines of curvature. If $k_2=0$, the $v$-curves are necessarily asymptotic curves (as well as lines of curvature).
