I know that if $\alpha$ is geodesic and its curvature is never zero, and it's plane, then it's a line of curvature (i.e. the tangent is a principal direction). I can prove this using Frenet.
I want to show first that all points are umbilical, because then I know how to prove that the curvature is constant, so the surface must be in a plane, a pshere, or the pseudosphere (but it can't be the pseudosphere because of reasons).
Given a point in the surface, and a direction, there exists one and only one geodesic in that direction. If the curvature is never zero, the direction is principal. If I can do this with all points and all directions, all points are umbilical.
But…it can happen that the curvature of the geodesic is zero and I don't know what to do in that case. Can you help me?
Best Answer
HINT: Suppose the geodesic has $\kappa(P) = 0$ for some $P$. Then $P$ must be a planar point, a parabolic point, or a hyperbolic point of the surface. In the latter two cases, you get $\kappa_n\ne 0$ for all but one or two directions, and you're done. Now what if $P$ is a planar point?
Now it's a set-theoretic argument. You need to assume the surface is connected, of course.