Search

A survey of works on discrete and continuous rigidity/nonrigidity and infinitesimal rigidity/nonrigidity of multidimensional surfaces, mainly in Euclidean spaces, is given. As a starting point for the methods of investigation, one considers three forms of the main theorem of the theory of surfaces (in local coordinates, in the invariant form, and in terms of exterior differential forms).

This article surveys results on the theory of deformation and infinitesimal deformation of surfaces in three-dimensional Euclidean space that have been obtained in the last 25–30 years and do not appear in previous survey articles.

Necessary and sufficient conditions are found such that the internally coalesced surface σ= S1 + S2 should have a parallel L ∃ S2 which divides the surface 2 into two parts so that the part σL, which does not contain a pole of the surface S2, should permit nontrivial bending slipping along L.

The nonrigidity of the composite surfaces of revolution 2 = S1+ S2+ S3 is analyzed, where S1 and S2 are internally glued and S2 and S3 are externally glued together. It is shown that cases of rigidity as well as nonrigidity can obtain for surfaces of this type.