New manifestations of the Darboux's rotation and translation fields of a surface

Abstract

We show how the rotation and translation fields of a surface, introduced by G. Darboux, may be used to obtain short proofs of a well-known theorem (that reads that the total mean curvature of a surface is stationary under an infinitesimal bending) and a new theorem (that reads that every infinitesimal flex of any simply connected closed surface is orthogonal to the surface at least at two points).