Representation of a smooth isometric deformation of a planar material region into a curved surface (Q1695380)

The authors provide necessary and sufficient conditions, as well as the corresponding coordinate representations, for the smooth (\({\mathcal C}^3\)-class) isometric deformations of a planar material region (identified with an open connected subset \({\mathcal D}\subset {\mathbb E}^2\)) into a surface \({\mathcal S}\subset {\mathbb E}^3\). Kinematic aspects of the two-dimensional continuous material regions (stretches of material filaments, distortion of material reference configuration) impose a stronger definition of isometric deformation than the classic one that is used in differential geometry. A strategy for determining such kinematic isometric deformations is presented together with explicit examples of the corresponding representations.