Soit un ouvert connexe et simplement connexe de tel que la distance géodésique dans soit équivalente à la distance euclidienne. Soit (gij) une métrique riemannienne de classe et de courbure nulle dans , telle que les fonctions gij et leurs dérivées partielles d'ordre aient des extensions continues à . Alors il existe un ouvert connexe de contenant et une métrique riemannienne de classe et de courbure nulle dans qui prolonge la métrique (gij).
Let be a connected and simply-connected open subset of such that the geodesic distance in is equivalent to the Euclidean distance. Let there be given a Riemannian metric (gij) of class and of vanishing curvature in , such that the functions gij and their partial derivatives of order have continuous extensions to . Then there exists a connected open subset of containing and a Riemannian metric of class and of vanishing curvature in that extends the metric (gij).
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@article{CRMATH_2004__338_5_391_0, author = {Ciarlet, Philippe G. and Mardare, Cristinel}, 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}, url = {http://www.numdam.org/articles/10.1016/j.crma.2003.12.017/} }
TY - JOUR AU - Ciarlet, Philippe G. AU - Mardare, Cristinel TI - Extension of a Riemannian metric with vanishing curvature JO - Comptes Rendus. Mathématique PY - 2004 SP - 391 EP - 396 VL - 338 IS - 5 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2003.12.017/ DO - 10.1016/j.crma.2003.12.017 LA - en ID - CRMATH_2004__338_5_391_0 ER -
%0 Journal Article %A Ciarlet, Philippe G. %A Mardare, Cristinel %T Extension of a Riemannian metric with vanishing curvature %J Comptes Rendus. Mathématique %D 2004 %P 391-396 %V 338 %N 5 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2003.12.017/ %R 10.1016/j.crma.2003.12.017 %G en %F CRMATH_2004__338_5_391_0
Ciarlet, Philippe G.; Mardare, Cristinel. Extension of a Riemannian metric with vanishing curvature. Comptes Rendus. Mathématique, Tome 338 (2004) no. 5, pp. 391-396. doi : 10.1016/j.crma.2003.12.017. http://www.numdam.org/articles/10.1016/j.crma.2003.12.017/
[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] 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 , 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 , 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] Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc., Volume 36 (1934), pp. 63-89
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