Let $M$ be a smooth orientable surface in $\mathbb{R}^{3}$, and $N$ a unit normal vector field along $M$. Assume that the surface has no planar point, i.e., for any $p$ in $M$, the second fundamental form is non-vanishing.
If $\gamma$ is a smooth regular curve in $M$, I believe the following is true:
If $N$ is constant along $\gamma$, then $\gamma$ is a straight line segment.
How would you prove my claim?