Constant normal vector along surface curve implies straight line 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?

This claim is false. Just consider a torus sitting on a plane. The point of intersection of the torus and the plane is an honest circle which is certainly not a straight line. However the normal to the surface is constant along this curve. 
