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

Ciarlet, Philippe G.; Mardare, Cristinel

Comptes Rendus. Mathématique, Tome 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 $\Omega \subset {ℝ}^n$, 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 $\Omega$, this classical result can be extended “up to the boundary”. When $\Omega$ 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 ${𝒞}^{\ell }\left(\overline{\Omega }\right)$.

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 $\Omega \subset {ℝ}^n$ 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 $\Omega$, comment ce résultat classique peut être étendu “jusqu'à la frontière”. Lorsque $\Omega$ 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 ${𝒞}^{\ell }\left(\overline{\Omega }\right)$.

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