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I know I can use Theorema Egregium to determine if a surface DNE. However using this method, I can convince myself that I can prove exisitence.

For example, I'm now working with a question given $E=1$, $F=0$, $G=\cos^2u$ in First Fundamental Form and $e=\cos^2 u$, $f=0$, $g=1$ in the Second Fundamental Form. I figured out a guess surface

$$x(u,v)=(\cos u \cos v, \cos u \sin v, \sin u)$$

such that $E,F,G,K$ meets the requirement (for $0<u<\pi/2$, $0<v<\pi/2$), but $e=1$, $f=0$, $g=\cos^2 u$.

If I'm understanding correctly, Theorema Egregium is not saying the other way round. So it's a "likely", not "existence". Also by the strict criteria to meet the requirement it's hard to imagine having a different surface for me. Instinct tells me that the values of $e$ and $g$ are indeed flipped, but I can't prove.

Please help. Thanks.

MonkeyKing
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    You are essentially asking for the compatibility equations between the first and second fundamental forms. These are given by Gauss' equation and the Codazzi-Mainardi Equations. The corresponding theorem is something along the following lines: Given a first and second fundamental, $I$ and $II$, respectively, defined on a convex open set U in the plane, then there is an immersion $f : U \to \mathbb{R}^{3}$ that realizes $I$ and $II$ as its first and second fundamental form if and only if $I$ and $II$ satisfy Gauss' equation and the Codazzi-Mainardi equations. – THW Nov 11 '16 at 19:03

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