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Euclidean isometries in $\mathbb{R}^3$ are compositions of a translation and an orthogonal transformation. Each Euclidean isometry is a surface isometry that preserves length of rectifiable curves, but the converse is not true. A length-preserving isometry between $\mathbb{R}^3$ surfaces in $S_1$ and $S_2$ is a bijection that preserves dot products of tangent vectors.

For local isometries there is plenty of example, but I need an example of surface isometry that is not an Euclidean isometry.

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Any example that you can imagine making of paper will do. Paper doesn't stretch or shrink, it only bends, so if you can turn some paper surface into another paper surface, they're related by a surface isometry. For instance, if you have an infinite cylinder, you can squash it so as to make the cross section elliptical, which obviously changes the distances of the points in $\mathbb R^3$, but doesn't change their distances on the surface.

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But the sheet of paper transformations are local isometries, not global isometries, right? (Global) isometries should be homeomorphism too, and obviously a rectangle in the plane is not homeoomorphic with a cylinder... but probably I am a bit confused. – user14174 Sep 24 '11 at 7:48
@Lmn6: I'm not sure I understand what you mean. A rectangle in a plane locally has the same metric as a cylinder, but globally it has a different topology. You don't get a paper cylinder from a paper rectangle just by bending the paper; you also have to join two edges. In my example of the circular and elliptical cylinders, there's no joining of edges, and the isometry is global. – joriki Sep 24 '11 at 8:03
ok, I don't know why I was supposing the paper was bent and the edges were joined together... Thank you. – user14174 Sep 24 '11 at 12:31

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