On the recovery of a manifold with boundary in $ \mathbb{R}^{n}$

Philippe G. Ciarlet; Cristinel Mardare

doi:10.1016/j.crma.2003.12.018

Mathematical Problems in Mechanics/Differential Geometry

On the recovery of a manifold with boundary in

ℝ^{n}

Presented by: Robert Dautray

Philippe G. Ciarlet ¹; Cristinel Mardare 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

Comptes Rendus. Mathématique, Volume 338 (2004) no. 4, pp. 333-340.

Abstract
Résumé

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 $Ω \subset ℝ^{n}$ , then $𝐂$ is the metric tensor field of a manifold isometrically immersed in $ℝ^{n}$ .

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 $𝒞^{ℓ} (\bar{Ω})$ .

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 $Ω \subset ℝ^{n}$ simplement connexe, alors $𝐂$ est le champ de tenseurs métriques d'une variété plongée isométriquement dans $ℝ^{n}$ .

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 $𝒞^{ℓ} (\bar{Ω})$ .

Received: 2003-12-01
Accepted: 2003-12-15
Published online: 2004-01-24

DOI: 10.1016/j.crma.2003.12.018

Author's affiliations:

Philippe G. Ciarlet ¹; Cristinel Mardare 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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     title = {On the recovery of a manifold with boundary in $ \mathbb{R}^{n}$},
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Philippe G. Ciarlet; Cristinel Mardare. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.12.018/

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