On rigid and infinitesimal rigid displacements in shell theory

Let ω be an open connected subset of ℝ² and let θ be an immersion from ω into ℝ³. It is first established that the set formed by all rigid displacements, i.e., that preserve the metric and the curvature, of the surface θ (ω) is a submanifold of dimension 6 and of class C^∞ of the space H¹(ω). It is then shown that the vector space formed by all the infinitesimal rigid displacements of the surface θ(ω) is nothing but the tangent space at the origin to this submanifold. In this fashion, the "infinitesimal rigid displacement lemma on a surface", which plays a key role in shell theory, is put in its proper perspective.