Extension of a Riemannian metric with vanishing curvature

Philippe G. Ciarlet; Cristinel Mardare

doi:10.1016/j.crma.2003.12.017

Differential Geometry

Extension of a Riemannian metric with vanishing curvature
[Prolongement d'une métrique riemannienne à courbure nulle]

Présenté par : 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. 5, pp. 391-396.

Résumé
Abstract

Soit $Ω$ un ouvert connexe et simplement connexe de $ℝ^{n}$ tel que la distance géodésique dans $Ω$ soit équivalente à la distance euclidienne. Soit (g_ij) une métrique riemannienne de classe $𝒞^{2}$ et de courbure nulle dans $Ω$ , telle que les fonctions g_ij et leurs dérivées partielles d'ordre $⩽ 2$ aient des extensions continues à $\bar{Ω}$ . Alors il existe un ouvert connexe $\tilde{Ω}$ de $ℝ^{n}$ contenant $\bar{Ω}$ et une métrique riemannienne $({\tilde{g}}_{ij})$ de classe $𝒞^{2}$ et de courbure nulle dans $\tilde{Ω}$ qui prolonge la métrique (g_ij).

Let $Ω$ be a connected and simply-connected open subset of $ℝ^{n}$ such that the geodesic distance in $Ω$ is equivalent to the Euclidean distance. Let there be given a Riemannian metric (g_ij) of class $𝒞^{2}$ and of vanishing curvature in $Ω$ , such that the functions g_ij and their partial derivatives of order $⩽ 2$ have continuous extensions to $\bar{Ω}$ . Then there exists a connected open subset $\tilde{Ω}$ of $ℝ^{n}$ containing $\bar{Ω}$ and a Riemannian metric $({\tilde{g}}_{ij})$ of class $𝒞^{2}$ and of vanishing curvature in $\tilde{Ω}$ that extends the metric (g_ij).

Reçu le : 2003-12-01
Accepté le : 2003-12-15
Publié le : 2004-01-29

DOI : 10.1016/j.crma.2003.12.017

Affiliations des auteurs :

Philippe G. Ciarlet ¹ ; Cristinel Mardare ²

@article{CRMATH_2004__338_5_391_0,
     author = {Philippe G. Ciarlet and Cristinel Mardare},
     title = {Extension of a {Riemannian} metric with vanishing curvature},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {391--396},
     publisher = {Elsevier},
     volume = {338},
     number = {5},
     year = {2004},
     doi = {10.1016/j.crma.2003.12.017},
     language = {en},
}

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%A Cristinel Mardare
%T Extension of a Riemannian metric with vanishing curvature
%J Comptes Rendus. Mathématique
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Philippe G. Ciarlet; Cristinel Mardare. Extension of a Riemannian metric with vanishing curvature. Comptes Rendus. Mathématique, Volume 338 (2004) no. 5, pp. 391-396. doi : 10.1016/j.crma.2003.12.017. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.12.017/

Bibliographie
Cité par

[1] S. Anicic, H. Le Dret, A. Raoult, The infinitesimal rigid displacement lemma in Lipschitz coordinates and application to shells with minimal regularity, in press

[2] P.G. Ciarlet; F. Larsonneur On the recovery of a surface with prescribed first and second fundamental forms, J. Math. Pures Appl., Volume 81 (2002), pp. 167-185

[3] P.G. Ciarlet, C. Mardare, On the recovery of a manifold with boundary in $ℝ^{n}$ , C. R. Acad. Sci. Paris, Ser. I, in press

[4] P.G. Ciarlet, C. Mardare, Recovery of a manifold with boundary and its continuity as a function of its metric tensor, in press

[5] P.G. Ciarlet, C. Mardare, Extension of a surface in $ℝ^{3}$ , in press

[6] P.G. Ciarlet, C. Mardare, Recovery of a surface with boundary and its continuity as a function of its two fundamental forms, in press

[7] H. Whitney Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc., Volume 36 (1934), pp. 63-89

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