Isometric surfaces

Surfaces in a Euclidean or Riemannian space for which there is a one-to-one correspondence between their points under which every rectifiable curve in one surface corresponds to a rectifiable curve of the same length in the other. In other words, isometric surfaces are characterized by (pairwise) isometric correspondences — isometries (see Isometric mapping) relative to the interior metrics (cf. Internal metric) induced on them by the metric of the ambient space. The most important example of isometric surfaces is the family of surfaces obtained by isometric deformation (cf. Deformation, isometric) of a given surface.

If the isometry of surfaces implies congruence, more exactly, if any surface $ F $ in some class $ K $ isometric to a surface $ F _ {0} $ has the property that the isometry is necessarily represented by a restriction of a self-isometry of the ambient space, then $ F _ {0} $ is said to be uniquely determined or rigid in the class $ K $.

The concept of isometric surfaces generalizes to wider categories of metric spaces and their subsets.

References

[a1]	R.S. Millman, G.D. Parker, "Elements of differential geometry" , Prentice-Hall (1977) pp. Sects. 4–10
[a2]	A.V. Pogorelov, "Extrinsic geometry of convex surfaces" , Amer. Math. Soc. (1973) (Translated from Russian)