Coefficients of first and second fundamental form if gaussian and mean curvatures are constant. I was solving a problem when at some point, I had this question. If $K$ and $H$ are constant for a surface. Can I say something about the coefficients of first and second fundamental form? I know that 
$K=\frac {LN-M^2} {EG-F^2}$  and $H=\frac{EN+GL-2FM} {2(EG-F^2)}$
So, for example, I have $K=0$ assuming $k_1>k_2$ and i know that $k_2=0$, I will get the correct coefficients if I put $E=1, F=0, G=1, L=k_1, M=0, N=0$ but is there a way to prove that this is the unique choice or a way to find them just by using that $K,H$ are constant?
Any help will be greatly appreciated.
 A: This is certainly not the unique choice in general. In the case $K = 0$, the surface is locally isometric to the plane. So, locally one can choose Euclidean coordinates for which $E = G = 1$, $F = 0$. Like John writes, one can produce other local coordinates by composing with some diffeomorphism, and produce a wide variety of triples $E, F, G$:
For example, one can easily find coordinates for which $E, F, G$ are any prescribed constants such that $EG - F^2 > 0$. Likewise, one can locally choose polar coordinates $(r, \theta)$, for which $E = 1$, $F = 0$, $G = r^2$.
On the other hand, one can produce conditions on $E, F, G, L, M, N$ beyond the algebraic equations for $K, H$ in terms of those components: In coordinates $(u, v)$, they satisfy the Gauss-Codazzi equations, which are first-order differential conditions:
\begin{align}
L_v - M_u &= L \Gamma^1_{12} + M(\Gamma^2_{12} - \Gamma^1_{11}) - N \Gamma^2_{11} \\
M_v - N_u &= L \Gamma^1_{22} + M(\Gamma^2_{22} - \Gamma^1_{12}) - N \Gamma^2_{12} \textrm{;}
\end{align}
here, $\Gamma^c_{ab}$ are the usual Christoffel symbols, which themselves are functions of $E, F, G$, and their first derivatives.
