Ordinary Differential Equations
On the recovery of a curve isometrically immersed in a Euclidean space
Comptes Rendus. Mathématique, Volume 338 (2004) no. 6, pp. 447-452.

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 𝒞 .

Il est connu en géométrie différentielle que l'on peut reconstruire une courbe à partir de ses n−1 fonctions de courbure, si l'on peut dériver ces fonctions suffisamment de fois dans le sens classique et si les premières n−2 fonctions sont strictement positives. On montre ici que ce résultat reste vrai sous l'hypothèse que les fonctions de courbure appartiennent à des espaces de Sobolev, en utilisant la notion de dérivée au sens des distributions. On montre aussi que l'application qui associe à ces fonctions de courbure la courbe ainsi construite est de classe 𝒞 .

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2004.01.018
Szopos, Marcela 1

1 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 4, place Jussieu, 75005 Paris, France
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Szopos, Marcela. On the recovery of a curve isometrically immersed in a Euclidean space. Comptes Rendus. Mathématique, Volume 338 (2004) no. 6, pp. 447-452. doi : 10.1016/j.crma.2004.01.018. http://www.numdam.org/articles/10.1016/j.crma.2004.01.018/

[1] Ciarlet, P.G. On the continuity of a surface as a function of its two fundamental forms, J. Math. Pures Appl., Volume 82 (2003), pp. 253-274

[2] Ciarlet, P.G.; Laurent, F. Continuity of a deformation as a function of its Cauchy–Green tensor, Arch. Rational Mech. Anal., Volume 167 (2003), pp. 255-269

[3] Klingenberg, W. A Course in Differential Geometry, Springer-Verlag, 1978

[4] Lions, J.-L.; Magenes, E. Problèmes aux limites non homogènes et applications, vol. I, Dunod, Paris, 1968

[5] Mardare, S. On isometric immersions of a Riemannian space under weak regularity assumptions, C. R. Acad. Sci. Paris, Ser. I, Volume 337 (2003), pp. 785-790

[6] S. Mardare, On the fundamental theorem of surface theory under weak regularity assumptions, C. R. Acad. Sci. Paris, Ser. I, in press

[7] M. Szopos, On the recovery of a curve isometrically immersed in a Euclidean space, in press

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