Differential Geometry
Extension of a Riemannian metric with vanishing curvature
Comptes Rendus. Mathématique, Volume 338 (2004) no. 5, pp. 391-396.

Let Ω be a connected and simply-connected open subset of n such that the geodesic distance in Ω is equivalent to the Euclidean distance. Let there be given a Riemannian metric (gij) of class 𝒞 2 and of vanishing curvature in Ω, such that the functions gij and their partial derivatives of order 2 have continuous extensions to Ω ¯. Then there exists a connected open subset Ω ˜ of n containing Ω ¯ and a Riemannian metric (g ˜ ij ) of class 𝒞 2 and of vanishing curvature in Ω ˜ that extends the metric (gij).

Soit Ω un ouvert connexe et simplement connexe de n tel que la distance géodésique dans Ω soit équivalente à la distance euclidienne. Soit (gij) une métrique riemannienne de classe 𝒞 2 et de courbure nulle dans Ω, telle que les fonctions gij et leurs dérivées partielles d'ordre 2 aient des extensions continues à Ω ¯. Alors il existe un ouvert connexe Ω ˜ de n contenant Ω ¯ et une métrique riemannienne (g ˜ ij ) de classe 𝒞 2 et de courbure nulle dans Ω ˜ qui prolonge la métrique (gij).

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2003.12.017
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
@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, Volume 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] Ciarlet, P.G.; Larsonneur, F. 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 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 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] Whitney, H. Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc., Volume 36 (1934), pp. 63-89

Cited by Sources: