Question: Geodesics in a connected surface S parametrized by arc length with constant binormal vector implies that S are open of planes and spheres?
My idea: I know that binormal vector is defined by $b=t\times n=\alpha '\times\frac{\alpha ''}{k}$.
On the other hand, it is true that if all geodesics in a connected surface are plane curves, then this surface is a subset of a plane or a sphere. I know how to prove this too. Thus, I'm trying to find a way to connect the question above with this result.
However I'm not so sure if this is the best idea. I even do not know if the question above is true because I didn't find any counterexample. Would you help me with this?