I want to do it in 3 steps:
1) If a geodesic is a plane curve, so it is a line of curvature.
2) If all curves in a surface are line of curvatures, so all points of $S$ are umbilical.
3) If all points in a surface are umbilical, so the surface is contained in a sphere or a plane.
Well, the steps 1) and 3) are done, but I really don't know how to do the step 2).
I know the definition of umbilical points. It is where $\kappa_{1}=\kappa_{2}$, where $\kappa_{i}$ are the principal curvatures. How can I prove this if the point is contained in a geodesic plane curve (and also a line of curvature, by 1))?
Best Answer
Given a point $p$ in $S$ and $w(\neq0) \in T_pS$ there exists an $\epsilon > 0$ and a unique parametrized geodesic curve $\gamma \colon (-\epsilon,\epsilon) \rightarrow S$ such that $\gamma(0)=p$ and $\gamma'(0)=w$. (If you are referring Do Carmo it's in sec. 4.4 prop. 4)
So for a point $p$ in $S$ and for all $w \in T_pS$ there exists a unique geodesic curve which is a principal curve. I think you can draw conclusion from here.