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Comptes Rendus Mathématique
Volume 337, n° 12
pages 785-790 (décembre 2003)
Doi : 10.1016/j.crma.2003.09.039
Received : 22 September 2003 ;  accepted : 28 September 2003
On isometric immersions of a Riemannian space under weak regularity assumptions

Sorin  Mardare
Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 4, place Jussieu, 75005 Paris, France 

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We consider a Riemannian metric in an open subset of   and assume that its Riemann curvature tensor vanishes. If the metric is of class  , a classical theorem in differential geometry asserts that the Riemannian space is locally isometrically immersed in the  -dimensional Euclidean space. We establish that, if the metric belongs to the Sobolev space   and its Riemann curvature tensor vanishes in the space of distributions, then the Riemannian space is still locally isometrically immersed in the  -dimensional Euclidean space. To cite this article: S. Mardare, C. R. Acad. Sci. Paris, Ser. I 337 (2003).

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On considère une métrique Riemannienne dans un ouvert de   et on suppose que son tenseur de courbure de Riemann s'annule. Si la métrique est de classe  , un théorème classique en géométrie différentielle affirme que l'espace de Riemann peut être plongé localement dans l'espace euclidien  -dimensionnel par une immersion isométrique. On établit que, si la métrique est de classe   et son tenseur de courbure de Riemann s'annule, alors l'espace de Riemann peut encore être plongé localement dans l'espace euclidien  -dimensionnel par une immersion isométrique. Pour citer cet article : S. Mardare, C. R. Acad. Sci. Paris, Ser. I 337 (2003).




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