Let $S$ be a surface, and $p$ a point on $S$ with zero Gaussian curvature. Can there exist two curves $\gamma_1, \gamma_2 : (a,b) \to S$ with $\gamma_i(0)=p$ and parametrized by arc length, such that their normal curvatures have opposite signs at time $t=0$?
I want to know if this is correct:
According to the Euler formula, the normal curvature of a curve $\gamma$ at time $t=0$ is equal to $k_1\cos^2(\theta) + k_2\sin^2(\theta)$, where $\theta$ is the angle that $\gamma'(0)$ forms with $w_1$, one of the two vectors generating the tangent plane to the surface at $p$ (the other is $w_2$), and $k_1, k_2$ are the two principal curvatures. Remembering that $k_1 \cdot k_2$ is equal to the Gaussian curvature, then either $k_1=k_2=0$ or one of $k_1$ and $k_2$ is $0$. In the first case, the normal curvature becomes zero; in the second case, the normal curvature is either always positive or always negative, so they cannot have opposite signs.