On the recovery of a curve isometrically immersed in a Euclidean space

It is known from differential geometry that one can reconstruct a curve with n?1 prescribed curvature functions, if these functions can be differentiated a certain number of times in the usual sense and if the first n?2 functions are strictly positive. We establish here that this result still holds under the assumption that the curvature functions belong to some Sobolev spaces, by using the notion of derivative in the distributional sense. We also show that the mapping that associates with such prescribed curvature functions the reconstructed curve is of class $\text{C}{}^{\mathrm{\infty }}$. To cite this article: M. Szopos, C. R. Acad. Sci. Paris, Ser. I 338 (2004).