Suppose I have a bounded surface with principle curvatures = $0$. I am given the first fundamental form $\mathbf{a}$ with entries $E, F, G$. I know that these entries are obtained by taking some mapping $\vec{m}(u,v)$ and dotting its various derivatives, eg. $E = \vec{m}_u \cdot \vec{m}_u$. Is it possible to work backwards from $\mathbf{a}$ to obtain a unique $\vec{m}$?
For example I could construct a system of equations based on these derivative components:
$$ m_{u1}(m_{u1}) + m_{u2}(m_{u2}) = E \\ m_{u1}(m_{v1}) + m_{u2}(m_{v2}) = F\\ m_{v1}(m_{v1}) + m_{v2}(m_{v2}) = G$$
Then if I had one extra equation I could solve for $m_{u1}, m_{v1}, m_{u2}, m_{v2}$, and integrate each term to recover the original $\vec{m}(u,v)$. I suspect that the extra equation I need in this case might be the first Gauss-Codazzi equation (relating the entries of $\mathbf{a}$ to $K$).
Thanks!
If the second fundamental form is identically $0$, then it's a one-line proof that (assuming the surface is connected) the surface is a subset of a plane. You cannot tell which plane — as I said before, any rigid motion preserves first and second fundamental forms.