# Prove that if all geodesics of a surface $S$ are planar curves, then $S$ is contained in a plane or a sphere

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?

• oh, never mind. If the curvature is zero, then as $\alpha\ \cdot N = 0$, deriving that I get that the direction is principal and that the eigenvalue is zero! – Guillermo Mosse Nov 28 '16 at 19:08
• can u please explain the steps u take? – Eduardo Silva Jan 31 '18 at 1:00
• If you are still interested, I will think of it in a few days. Sorry that I can't now – Guillermo Mosse Feb 2 '18 at 12:32

## 1 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.