Let $\Sigma^2 \subseteq \mathbb{R}^3$ be a complete minimal surface and let's assume that there exists a smooth regular curve $\gamma \colon I \to \Sigma$ such that $K(\gamma(t)) = 0$ for all $t \in I$, where $K$ is the Gauss curvature of $\Sigma$.
Can I conclude that $\Sigma$ is flat, i.e. a plane?