On isometric immersions of a Riemannian space under weak regularity assumptions

Abstract

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 C 2, 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 W 1,∞ 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. To cite this article: S. Mardare, C. R. Acad. Sci. Paris, Ser. I 337 (2003).