View source for 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|Isometric mapping]]) relative to the interior metrics (cf. [[Internal metric|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|Deformation, isometric]]) of a given surface.

If the isometry of surfaces implies congruence, more exactly, if any surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052830/i0528301.png" /> in some class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052830/i0528302.png" /> isometric to a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052830/i0528303.png" /> has the property that the isometry is necessarily represented by a restriction of a self-isometry of the ambient space, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052830/i0528304.png" /> is said to be uniquely determined or rigid in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052830/i0528305.png" />.

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



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