Differential Geometry general question

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If the coefficients of the First Fundamental Form differ for two parameterizations, it does not automatically indicate that the surfaces are not isometric. The discussion centers on proving that two surfaces with different FFF coefficients but the same Gaussian curvature are not isometric, challenging the converse of Gauss's Great Theorem. The user seeks assistance in demonstrating this non-isometry despite the shared Gaussian curvature. Understanding the implications of the First Fundamental Form and Gaussian curvature is crucial for this proof. The conversation highlights the complexities in differential geometry regarding surface parameterizations and isometry.
InbredDummy
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Ok, in general, I know that if the coefficients of the First Fundamental Form agree for two surfaces parameterized by X and Y, the the map X(Y^-1) is an isometry, or the two surfaces are isometric.

I also know that if two parameterizations don't have the same coefficients, this does not imply that the two surfaces are not isometric.

So i have two parameterizationsthat have different coefficients of the FFF (first fundamental form) but have the same Gaussian curvature. I need to prove that the two surfaces are not isometric. (ie i need to prove that the converse of Gauss's Great Theorem is false).
 
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