On rigid displacements and their relation to the infinitesimal rigid displacement lemma in shell theory

Ciarlet, Philippe G.; Mardare, Cristinel

Comptes Rendus. Mathématique, Tome 336 (2003) no. 11, pp. 959-966.

Soit ω un ouvert connexe de ${ℝ}^2$ et $\theta$ une immersion de ω dans ${ℝ}^3$. On établit que l'ensemble formé par les déplacements rigides de la surface $\theta \left(\omega \right)$ est une sous-variété de dimension 6 et de classe ${𝒞}^{\infty }$ de l'espace ${𝐇}^1\left(\omega \right)$. On montre aussi que les déplacements rigides infinitésimaux de la même surface $\theta \left(\omega \right)$ engendrent le plan tangent à l'origine à cette sous-variété.

Let ω be an open connected subset of ${ℝ}^2$ and let $\theta$ be an immersion from ω into ${ℝ}^3$. It is established that the set formed by all rigid displacements of the surface $\theta \left(\omega \right)$ is a submanifold of dimension 6 and of class ${𝒞}^{\infty }$ of the space ${𝐇}^1\left(\omega \right)$. It is shown that the infinitesimal rigid displacements of the same surface $\theta \left(\omega \right)$ span the tangent space at the origin to this submanifold.

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