[Justification de la condition de compatibilité de Darboux–Vallée–Fortuné en théorie des surfaces]
Etant donné deux champs suffisamment réguliers définis dans un ouvert simplement connexe , l'un de matrices symétriques définies positives et l'autre de matrices symétriques, le théorème fondamental de la théorie des surfaces affirme que, si ces deux champs satisfont les relations de Gauss et Codazzi–Mainardi dans ω, alors il existe une immersion θ de ω dans telle que ces champs soient les deux formes fondamentales de la surface .
On montre ici qu'une nouvelle relation de compatibilité, dont C. Vallée et D. Fortuné ont montré en 1996 la nécessité en suivant une idée de G. Darboux, est également suffisante pour l'existence d'une telle immersion θ.
Given two fields of positive definite symmetric, and symmetric, matrices defined over a simply-connected open subset , the fundamental theorem of surface theory asserts that, if these fields satisfy the Gauss and Codazzi–Mainardi relations in ω, then there exists an immersion θ from ω into such that these fields are the two fundamental forms of the surface
We show here that a new compatibility relation, shown to be necessary by C. Vallée and D. Fortuné in 1996 through the introduction, following an idea of G. Darboux, of a rotation field on a surface, is also sufficient for the existence of such an immersion θ.
Publié le :
@article{CRMATH_2008__346_21-22_1197_0,
author = {Ciarlet, Philippe G. and Iosifescu, Oana},
title = {Justification of the {Darboux{\textendash}Vall\'ee{\textendash}Fortun\'e} compatibility relation in the theory of surfaces},
journal = {Comptes Rendus. Math\'ematique},
pages = {1197--1202},
publisher = {Elsevier},
volume = {346},
number = {21-22},
year = {2008},
doi = {10.1016/j.crma.2008.09.002},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2008.09.002/}
}
TY - JOUR AU - Ciarlet, Philippe G. AU - Iosifescu, Oana TI - Justification of the Darboux–Vallée–Fortuné compatibility relation in the theory of surfaces JO - Comptes Rendus. Mathématique PY - 2008 SP - 1197 EP - 1202 VL - 346 IS - 21-22 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2008.09.002/ DO - 10.1016/j.crma.2008.09.002 LA - en ID - CRMATH_2008__346_21-22_1197_0 ER -
%0 Journal Article %A Ciarlet, Philippe G. %A Iosifescu, Oana %T Justification of the Darboux–Vallée–Fortuné compatibility relation in the theory of surfaces %J Comptes Rendus. Mathématique %D 2008 %P 1197-1202 %V 346 %N 21-22 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2008.09.002/ %R 10.1016/j.crma.2008.09.002 %G en %F CRMATH_2008__346_21-22_1197_0
Ciarlet, Philippe G.; Iosifescu, Oana. Justification of the Darboux–Vallée–Fortuné compatibility relation in the theory of surfaces. Comptes Rendus. Mathématique, Tome 346 (2008) no. 21-22, pp. 1197-1202. doi : 10.1016/j.crma.2008.09.002. http://www.numdam.org/articles/10.1016/j.crma.2008.09.002/
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