Following is a question from O'Neill's Elementary Differential Geometry: I want to determine whether the following curve is one or more of: principal, asymptotic, geodesic:
The x axis in $M:z=xy$.
The trouble I'm having is primarily the language. What does "x axis in a surface" mean? Is it the intersection of some plane with the surface that yields the x axis? If so, then the curve is simply a straight line in $\mathbb{R^3}$ and is asymptotic and principal.
The $x$-axis is the curve $y=z=0$ contained in the surface. It is a straight line, of course, contained in the surface and so is an asymptotic curve. Is it a geodesic? Is it a line of curvature (i.e., are its tangent vectors principal directions)? What do you know about the curvature of the saddle surface?