# 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?

• The question is, how would you prove your claim? – Edu Aug 4 '17 at 10:06
• What makes you believe that your claim is correct? I presume you've played around with lots and lots of examples? – Ted Shifrin Aug 4 '17 at 23:11