I am trying to solve this question, I understand the logic behind it which I will try to explain after the statement of the question. The question reads:
Let $S$ be an oriented regular surface that is tangent to the plane along a regular curve $\alpha$. Show that the points on $\alpha$ are parabolic or planar points of $S$.
Now considering the Normal Vector field $N$ along $\alpha$, we can see that this normal vector field is a normal vector field along the surface as well.
The plane has the same normal vector plane, and has a Gaussian Curvature of $0$. Thus because the surface has the same normal vector plane and is tangent to the plane there, shouldn't the Gaussian Curvature there also give $0$?
Is my logic sound or is there another way to prove this?
You are right. Take $p \in S$ and let $\alpha \colon ]-\epsilon , \epsilon[ \to \mathbb R^3$ be a unit speed parametrization of $\Gamma$. Since $\Gamma$ is planar we know that $\alpha$'s binormal is constant (zero torsion). Hence we may pick a patch such that the binormal and the surface normals agree along $\alpha$. Now define $v = \alpha'(o)$ then we have argued that $dN_p(v)=0$.
Let $B(p,\delta)$ be so small that the graph of $\alpha$ partition $S \cap B(p,\delta)$ in two parts. If the surface lies to the same side of $\Pi$ on each part then all other sectional curvatures are either non-negative or non-positive (depending on the side of $\Pi$). Therefore $v$ is a principal direction and the gaussian curvature is 0. If $S$ lies on different sides of $\Pi$ in our small $S \cap B(p,\delta)$ neighbourhood. Then the sectional curvatures are 0. And again $v$ is a principal direction. Thus the gaussian curvature is 0.