Comptes Rendus
no. 12
Differential Geometry/Mathematical Problems in Mechanics
On isometric immersions of a Riemannian space under weak regularity assumptions

Presented by: Philippe G. Ciarlet

Sorin Mardare1
1 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 4, place Jussieu, 75005 Paris, France
Comptes Rendus. Mathématique, Volume 337 (2003) no. 12, pp. 785-790.

We consider a Riemannian metric in an open subset of ℝ d and assume that its Riemann curvature tensor vanishes. If the metric is of class C2, a classical theorem in differential geometry asserts that the Riemannian space is locally isometrically immersed in the d-dimensional Euclidean space. We establish that, if the metric belongs to the Sobolev space W1,∞ and its Riemann curvature tensor vanishes in the space of distributions, then the Riemannian space is still locally isometrically immersed in the d-dimensional Euclidean space.

On considère une métrique Riemannienne dans un ouvert de ℝ d et on suppose que son tenseur de courbure de Riemann s'annule. Si la métrique est de classe C2, un théorème classique en géométrie différentielle affirme que l'espace de Riemann peut être plongé localement dans l'espace euclidien d-dimensionnel par une immersion isométrique. On établit que, si la métrique est de classe W1,∞ et son tenseur de courbure de Riemann s'annule, alors l'espace de Riemann peut encore être plongé localement dans l'espace euclidien d-dimensionnel par une immersion isométrique.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2003.09.039

Sorin Mardare 1

1 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 4, place Jussieu, 75005 Paris, France 
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Sorin Mardare. On isometric immersions of a Riemannian space under weak regularity assumptions. Comptes Rendus. Mathématique, Volume 337 (2003) no. 12, pp. 785-790. doi : 10.1016/j.crma.2003.09.039. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.09.039/

[1] 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

[2] L.C. Evans; R.F. Gariepy Measure Theory and Fine Properties of Functions, Stud. Adv. Math., CRC Press, Boca Raton, FL, 1992

[3] S. Mardare, On isometric immersions of a Riemannian space with little regularity, Analysis and Applications, in press

[4] V.G. Maz'ja Sobolev Spaces, Springer-Verlag, 1985

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