On rigid displacements and their relation to the infinitesimal rigid displacement lemma in three-dimensional elasticity

Let $\text{Ω}$ be an open connected subset of $\text{R}{}^3$ and let $\text{Θ}$ be an immersion from $\text{Ω}$ into $\text{R}{}^3$. It is established that the set formed by all rigid displacements of the open set $\text{Θ}\text{(Ω)}$ is a submanifold of dimension 6 and of class $\text{C}{}^{\mathrm{\infty }}$ of the space $\text{H}{}^1\text{(Ω)}$. It is also shown that the infinitesimal rigid displacements of the same set $\text{Θ}\text{(Ω)}$ span the tangent space at the origin to this submanifold. To cite this article: P.G. Ciarlet, C. Mardare, C. R. Acad. Sci. Paris, Ser. I 336 (2003).