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The term Isometric Surface brings up surprisingly unrelated search results on google. So I thought I would ask here. What is an isometric surface? I do have an understanding of what it is, but I need to confirm it as it may be completely wrong. If in your answer, you could give a brief comparison between them and implicit surfaces as well as parametric surfaces (I have a guess that isometric and parametric surfaces are similar/same... but I may be completely wrong), that would be great.

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closed as unclear what you're asking by Rahul, dfeuer, Vedran Šego, Nicholas R. Peterson, Cameron Buie Sep 22 '13 at 3:16

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If Robert Israel is right (which is to say, if the sun comes up tomorrow), there's no such thing as an isometric surface, and the rest of the question is a fishing expedition, so I've voted to close, pending some effort on Samaursa's part to put this in better shape. –  Gerry Myerson May 20 '11 at 5:51
    
Are you sure rather you mean Isoperimetric surface? –  Arjang May 20 '11 at 5:57
    
@GerryMyerson Okay, but there does seem to be a lot of people who think that they exist: encyclopediaofmath.org/index.php/Isometric_surfaces, google.ca/…. Plenty of research papers talk about this as well. This is an old question but I will revisit it once I run across those papers again. –  Samaursa Apr 9 at 16:00
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I think you are missing Robert Israel's point. There is no such thing as an isometric surface, much as there is no such thing as a similar triangle, or a parallel line. Just as parallel lines are two or more lines that are parallel to each other, with no one line in isolation being a parallel line, isometric surfaces are two or more surfaces that are isometric to each other, with no one surface in isolation being an isometric surface. –  Gerry Myerson Apr 10 at 0:38
    
@GerryMyerson I definitely missed the point and that makes perfect sense! The similar-triangle and parallel-line example you gave finally put it all together for me, thanks! –  Samaursa Apr 10 at 11:54

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Two surfaces (or, in general, metric spaces) are isometric (to each other) if there is a one-to-one correspondence between them that preserves distances. But AFAIK there is no such thing in mathematics as "an isometric surface" on its own, rather than in relation to some other surface.

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Often in Riemannian geometry, an isometry between two manifolds (surfaces) is defined to be a diffeomorphism which pulls back the metric tensor on one to the other. It is an easy exercise to prove that an isometry in this sense is one in the sense you wrote about. What's a bit harder is the converse, that an isometry in your sense is an isometry in the Riemannian geometry sense. –  Jason DeVito May 20 '11 at 5:18

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