Mathematical Problems in Mechanics/Differential Geometry
On the recovery of a manifold with boundary in n
Comptes Rendus. Mathématique, Volume 338 (2004) no. 4, pp. 333-340.

If the Riemann curvature tensor associated with a smooth field 𝐂 of positive-definite symmetric matrices of order n vanishes in a simply-connected open subset Ω n , then 𝐂 is the metric tensor field of a manifold isometrically immersed in n .

In this Note, we first show how, under a mild smoothness assumption on the boundary of Ω, this classical result can be extended “up to the boundary”. When Ω is bounded, we also establish the continuity of the manifold with boundary obtained in this fashion as a function of its metric tensor field, the topologies being those of the Banach spaces 𝒞 (Ω ¯).

Si le tenseur de courbure de Riemann associé à un champ régulier 𝐂 de matrices symétriques définies positives d'ordre n s'annule sur un ouvert Ω n simplement connexe, alors 𝐂 est le champ de tenseurs métriques d'une variété plongée isométriquement dans n .

Dans cette Note, on montre d'abord, moyennant une hypothèse peu restrictive sur la régularité de la frontière de Ω, comment ce résultat classique peut être étendu “jusqu'à la frontière”. Lorsque Ω est borné, on établit aussi la continuité de la variété à bord ainsi obtenue en fonction de son champ de tenseurs métriques, les topologies étant celles des espaces de Banach 𝒞 (Ω ¯).

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2003.12.018
Ciarlet, Philippe G. 1; Mardare, Cristinel 2

1 Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
2 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 4, place Jussieu, 75005 Paris, France
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Ciarlet, Philippe G.; Mardare, Cristinel. On the recovery of a manifold with boundary in $ \mathbb{R}^{n}$. Comptes Rendus. Mathématique, Volume 338 (2004) no. 4, pp. 333-340. doi : 10.1016/j.crma.2003.12.018. http://www.numdam.org/articles/10.1016/j.crma.2003.12.018/

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