On rigid and infinitesimal rigid displacements in three-dimensional elasticity

Let Omega be an open connected subset of R-3 and let Theta be an immersion from Omega into R-3. It is first established that the set formed by all rigid displacements, i.e. that preserve the metric, of the open set Theta(Omega) is a submanifold of dimension 6 and of class C-infinity of the space H-1(Omega). It is then shown that the vector space formed by all the infinitesimal rigid displacements of the same set Theta(Omega) is nothing but the tangent space at the origin to this submanifold. In this fashion, the familiar "infinitesimal rigid displacement lemma" of three-dimensional linearized elasticity is put in its proper perspective.