If two objects have the same gaussian curvature, are they the same up to isometries? I was reading about Gauss Egregium Theorem but I'm not sure if I understand it well. Intuitively, what does it mean? It is true that if two objects have the same Gaussian curvature, then they are the same, OR is true that if two objects have an isometry, then they have the same curvature?
The statement says that Gaussian curvature is preserved under isometries, but the trouble starts with the word preserves.
Any hint would be appreciated.
 A: If two surfaces have the same Gaussian curvature, it is not true that they are the same up to an isometry.
A counterexample for that is the exponential horn ($X_1(u,v) = (u \cos v, u \sin v, \log u)$) and the cylinder ($X_2(u,v) = (u \cos v, u \sin v, v)$), which have same Gaussian curvature at corresponding points, but are actually not isometric (calculate the first fundamental form and see that they are essentially different).
What the Theorem Egregium actually says is that, if two surfaces are locally isometric, then they MUST have same Gaussian curvature at corresponding points. See it another way: if two surfaces have different Gaussian curvatures at corresponding points under a map, then that map cannot be an isometry since, by the Theorem, the surfaces would have same Gaussian curvature.
A: Intuitively, the Theorema Egregium means that you can bend or shift a space, just so long as you don't stretch it, and the Gaussian curvature will remain the same. This means that it doesn't matter how you embed a manifold in space, the curvature will always be the same. You could also interpret this as: two objects which are isometric have the same curvature.
See here for some examples of objects have the same curvature, but which are not isometric.
A: Yes they are same upto local/superficial isometry, but have different Avatars so to say. You can, by a series of bending deformations distort any surface to "become"( except for Euclidean shifts/ rotations.. constants of integration) to some other surface with same areal distribution of $K$. Each triangular /rhombic differential element has same differential length and angle when bending.
When you just say "same" you mean re-locating a rigid object elsewhere... congruency.
Preserve, conserve, hold as invariant or constant .. they are all same, even if they seemingly appear with  different terminologies in vogue in surface theory.
" Bending " in common parlance is a word used in same connotation by notable authors and researchers in English, French and German.
For an example.. cut a plastic ball in two halves and bend one in your hands until it forms cones on each side. This one is a elliptic sphere, has locally same Gauss curvature $K$ everywhere, so long as you don't stretch or compress any part of this thin sheet/membrane. Of course in an air balloon a lot of stretch and $K$ decrease at all stretched points takes place, so the deformation is not isometric under pressure loading, the $K$ decreases visibly for inflated balloons.
To give a physics example: Just as in isothermy  of physical gases of given mass you have pressure and volume varying inversely ( constant product in Boyle's Law) and the physical constant product is chemically intrinsic related to the number of molecules present in  the gas which cannot either be added or destroyed..
so also in isometry the product of principal curvatures $ \kappa_1,\kappa_2  $ remains same, k1 varies inversely as k2 during the bending. The constant is "chemically" intrinsic, can be established isometrically in terms of  Christoffel symbols that are obtained purely from the first fundamental form. In Bending what changes is the ratio of principal curvatures$ \kappa_1 / \kappa_2,  $ and  normal curvatures.
A rectangular sheet of paper, a cylinder, cone, tangential (& polar) developables are shapes that can all be "inter-bent" from one shape to the other.
Recently I showed here  how to bend a rectangular sheet of paper to arbitrary cone angles.
and conversely if $K$ is conserved full isometry need not always exist.
EDIT1:
In Mechanics of Materials however, the mathematical concept of Bending as above is restricted to the neutral surface where all surface elements retain their lengths and angles during deformation. Elements of a Plate or Shell undergo linear strain and angular (shear) distortion at thin layer distances away from the neutral surface under the influence of Forces in the membrane and Bending Moments out of it. The Engineer's theory of Bending allows/includes strains in proportion to distance along surface normal. That is, $K$ need not remain the same in layers parallel to the neutral surface. This contradiction remains between pure mathematics and mechanics viewpoints because in the latter, reference is to overall strain behaviour everywhere and not just for locations in the neutral surface.
