If the first fundamental form of a surface is $\mathbb{I}$, then the Gauss-Peterson-Mainardi-Codazzi equations reduce to: $LN - M^2 = 0$, $\frac{\partial L}{\partial V} - \frac{\partial M}{\partial U} = 0$, and $\frac{\partial M}{\partial V} - \frac{\partial N}{\partial U} = 0$, where $L, M, N$ are the coefficients of the second fundamental form. $LN - M^2$ is also the Gaussian Curvature $K$, so the surface has $K = 0$ and is developable. However, I understood that the second fundamental form for any surface with Gauss curvature $0$ also has $M = N = 0$. To my eye, these conditions on $M$ and $N$ do not emerge from the GPMC equations. What am I missing?
Thank you!